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Sobolev Poincare and Morrey Inequalities

1 · Prerequisites

2 · Summary

This page develops the first-order Sobolev inequalities and the Poincare and Morrey inequalities on Euclidean domains. The Sobolev conjugate $p^{*} = n p / (n - p)$ is defined with its scaling identity and iterated form, and the endpoint $p = 1$ Gagliardo-Nirenberg-Sobolev inequality is proved by multiplying the $n$ one-dimensional primitive bounds. The pointwise potential bound for compactly supported smooth functions and the $W^{1, p}$ ball-mean oscillation estimate by the Riesz potential of the gradient provide the analytic engine: they are the substitutes for the maximal-function and $p > n$ arguments of the classical treatments, and they are proved from polar coordinates, Fubini, and the smooth approximation of Sobolev classes. The $1 < p < n$ inequality is obtained from the $p = 1$ case through the $∣ u ∣^{γ}$ device with $γ = p (n - 1) / (n - p)$ and Holder's inequality, first for compactly supported smooth functions and then for all of $W^{1, p} (R^{n})$ by density and completeness; the zero-boundary closure corollary follows by extension by zero.

The Poincare theory is developed for three domain classes. On a ball the convex-domain Poincare-Wirtinger corollary gives the mean-zero estimate with an explicit dimension-only constant linear in the radius. On bounded John domains the John curve is converted into a bounded-overlap chain of balls; telescoping the ball means along the chain, bounding the differences by the ball Poincare inequality, and summing with the truncated Riesz kernel bound gives the mean-zero inequality for every $1 \leq p < \infty$ . Bounded convex domains and the cone-condition classes are compared with the John class, and a positive-measure zero set normalisation is shown to control the mean, giving the zero-set Poincare inequality. On bounded connected extension domains the mean-zero estimate for $1 < p < n$ is proved by an extension-cutoff-mollification argument with Arzela-Ascoli, and the Sobolev-Poincare form at the critical exponent $p^{*}$ follows from the extension-domain embedding. On the whole space $W^{1, p}$ does not include into $L^{q}$ for $q < p$ , even with the full Sobolev norm; an explicit power tail witnesses the set-inclusion failure, while dilation disproves the continuous bound and the homogeneous Poincare estimate; the consequences are recorded on the examples page.

Morrey's inequality for $p > n$ is proved by combining the ball oscillation bound with Holder's inequality in the exponent $1 - n / p$ , comparing ball means at two scales, and identifying the continuous representative by Lebesgue differentiation; the same computation yields the local Holder bound with the norm taken on a fixed doubled ball. At $p = \infty$ the $W^{1, \infty}$ classes of a bounded convex domain are identified with Lipschitz classes through a multiplicity estimate for segment integrals, a.e.-pair Lipschitz bounds, Lebesgue points and the dense-subset extension theorem. The higher-order embedding iterates the first-order Sobolev and Morrey inequalities through the lower-order weak derivatives, covering the subcritical, critical and supercritical orders, and the Sobolev algebra property above the critical index follows from the higher-order Leibniz identity together with the iterated embeddings. Closing remarks record the $p = n$ endpoint, where the critical space embeds into every finite $L^{q}$ but not into $L^{\infty}$ , and the domain classes covered by the mean-zero Poincare inequality.

Conventions: $Ω \subseteq R^{n}$ is open, $n \geq 2$ , $K$ is $R$ or $C$ , and $∣ D u ∣$ denotes the Euclidean norm of the weak gradient. The Axiom of Choice is stated on the items whose proofs invoke the ACL, extension, density or Arzela-Ascoli interfaces; the mean-zero estimates are stated with their exact exponent ranges, and the constants depend only on the data named in each statement.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

The Sobolev conjugate exponent and the scaling identity

Definition

Let $n \geq 2$ and $1 \leq p < n$ . The Sobolev conjugate of $p$ is $p^{*} = \frac{n p}{n - p} .$ The denominator $n - p$ is positive, so $p^{*}$ is a finite real number greater than $1$ . Conjugate exponents in the sense of Conjugate exponents, including the endpoint conventions and real powers in the sense of Real powers for positive bases, with the zero-base positive-exponent convention are used below.

The following elementary identities are part of the definition and record how $p^{*}$ is used. $\frac{1}{p^{*}} = \frac{n - p}{n p} = \frac{1}{p} - \frac{1}{n}, p^{*} - p = \frac{n p - p (n - p)}{n - p} = \frac{p^{2}}{n - p} > 0,$ so $p^{*} > p$ and $1 / p^{*} + 1 / n = 1 / p$ ; in particular $p^{*} = n / (n - 1)$ when $p = 1$ . Equivalently $p^{*} = n / (\frac{n}{p} - 1)$ .

For $k \geq 1$ with $k p < n$ the $k$ -fold Sobolev conjugate $p_{k}^{*}$ is defined recursively by $\frac{1}{p_{1}^{*}} = \frac{1}{p} - \frac{1}{n}, \frac{1}{p_{j + 1}^{*}} = \frac{1}{p_{j}^{*}} - \frac{1}{n} (1 \leq j < k) .$ Induction on $j$ gives $\frac{1}{p_{j}^{*}} = \frac{1}{p} - \frac{j}{n}$ and hence $p_{k}^{*} = \frac{n p}{n - k p};$ the recursion is well posed because $j p < n$ for $j \leq k$ keeps every $p_{j}^{*}$ finite and greater than $1$ .

Finally, the map $p \mapsto p^{*} = n / (\frac{n}{p} - 1)$ is continuous on $(1, n)$ , since $n / p - 1 > 0$ there and the reciprocal and affine maps are continuous; it is strictly increasing on $(1, n)$ , because $p \mapsto n / p$ is strictly decreasing and $t \mapsto n / (t - 1)$ is strictly decreasing on $(1, \infty)$ ; and $p^{*} \to \infty$ as $p \to n^{-}$ , because $n / p - 1 \to 0^{+}$ . No finite Sobolev conjugate is attached to $p \geq n$ : for $p = n$ the formula has denominator $0$ , and for $p > n$ the expression $n p / (n - p)$ is negative.

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Pointwise potential bound for compactly supported smooth functions

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 2$ and $u \in C_{c}^{\infty} (R^{n}; K)$ , $K \in {R, C}$ . Then for every $x \in R^{n}$ , $∣ u (x) ∣ \leq \frac{1}{σ (S^{n - 1})} \int_{R^{n}} ∣ D u (y) ∣ ∣ x - y ∣^{1 - n} d y$ , where $σ$ is the polar surface measure of The polar surface set function on the unit sphere and $∣ D u ∣$ is the Euclidean norm of the gradient. In particular $∣ u (x) ∣ \leq C (n) \int_{R^{n}} ∣ D u (y) ∣ ∣ x - y ∣^{1 - n} d y$ .

Facts & Assumptions

Given: Countable Choice; an integer $n \geq 2$ ; a field $K \in {R, C}$ ; a function $u \in C_{c}^{\infty} (R^{n}; K)$ ; the polar surface measure $σ$ on $S^{n - 1}$ ; and a point $x \in R^{n}$ .

[F2]

Polar coordinates: for every Borel measurable $f : R^{n} \to [0, \infty]$ , $\int_{R^{n}} f d λ_{n} = \int_{0}^{\infty} \int_{S^{n - 1}} f (r ω) r^{n - 1} d σ (ω) d r$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

[F3]

Lebesgue measure is translation invariant (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).

[F4]

For a vector-valued differentiable $f$ with integrable derivative, $\int_{a}^{b} f^{'} = f (b) - f (a)$ (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz).

[F5]

Tonelli's theorem on sigma-finite products (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F6]

Every Euclidean ball has positive finite Lebesgue measure: $0 < λ (B (x, r)) < \infty$ (Euclidean balls have positive finite Lebesgue measure).

[F7]

Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1F4givenalgebra

The radial primitive. Fix $ω \in S^{n - 1}$ and put $g (t) : = u (x + t ω)$ for $t \geq 0$ . Since $u$ is smooth and compactly supported, $g$ is differentiable with $g^{'} (t) = D u (x + t ω) \cdot ω$ , and $g (t) = 0$ for all $t \geq T$ once $T$ is so large that $x + [0, \infty) ω$ leaves the support of $u$ . Applying the fundamental theorem [F4] on $[0, T]$ and letting $T \to \infty$ gives $g (0) = - \int_{0}^{\infty} g^{'} (t) d t$ , hence $∣ u (x) ∣ \leq \int_{0}^{\infty} ∣ D u (x + t ω) ∣ d t$ .

1.2F2F6algebra

Surface normalisation. By [F2] applied to $1_{B (0, 1)}$ , $λ_{n} (B (0, 1)) = σ (S^{n - 1}) \int_{0}^{1} t^{n - 1} d t = σ (S^{n - 1}) / n$ . Thus [F6] gives $0 < σ (S^{n - 1}) = n λ_{n} (B (0, 1)) < \infty$ .

1.3F2F3givenalgebra

Translation to polar coordinates at $x$ . Define $G (y) : = ∣ D u (y) ∣ ∣ x - y ∣^{1 - n}$ for $y \neq x$ and $G (x) : = 0$ ; this is Borel measurable because $∣ D u ∣$ is continuous and $y \mapsto ∣ x - y ∣^{1 - n}$ is Borel. Applying [F2] to the nonnegative Borel function $z \mapsto G (x + z)$ and then [F3] gives $\int_{S^{n - 1}} \int_{0}^{\infty} ∣ D u (x + t ω) ∣ d t d σ (ω) = \int_{S^{n - 1}} \int_{0}^{\infty} G (x + t ω) t^{n - 1} d t d σ (ω) = \int_{R^{n}} G (y) d y,$ where the first equality uses $∣ x - (x + t ω) ∣^{1 - n} = t^{1 - n}$ and the two integrations are the iterated polar integral of the nonnegative function $z \mapsto G (x + z)$ ; the singularity at $y = x$ is a single point and does not affect the value of the integral.

2.1F5F7step 1.1algebra

Integrating the pointwise bound over the sphere. The function $(t, ω) \mapsto ∣ D u (x + t ω) ∣$ is continuous on $[0, \infty) \times S^{n - 1}$ , hence product measurable, and it is nonnegative; by Tonelli [F5] its iterated integral over the sigma-finite product $[0, \infty) \times S^{n - 1}$ is well defined. Integrating the inequality of step 1.1 over $S^{n - 1}$ against $σ$ therefore gives $∣ u (x) ∣ σ (S^{n - 1}) \leq \int_{S^{n - 1}} \int_{0}^{\infty} ∣ D u (x + t ω) ∣ d t d σ (ω)$ .

3.1step 1.2step 2.1step 1.3algebra∎

Conclusion. Combining steps 2.1 and 1.3 with the positivity of $σ (S^{n - 1})$ from step 1.2 gives $∣ u (x) ∣ \leq \frac{1}{σ (S^{n - 1})} \int_{R^{n}} ∣ D u (y) ∣ ∣ x - y ∣^{1 - n} d y$ , and $C (n) : = 1 / σ (S^{n - 1}) = 1 / (n λ_{n} (B (0, 1)))$ is the asserted dimension-only constant.

Source notes

Kinnunen's Lemma 5.22 proves the corresponding oscillation bound on a ball by slicing spheres and changing variables; the proof above uses the same radial computation in the global polar-coordinate form suited to compactly supported functions, with the sphere average normalised by $σ (S^{n - 1})$ . Hunter's display (3.14) gives the related ball-averaged oscillation bound; the compact-support ray argument above gives the global estimate with $1 / σ (S^{n - 1})$ .

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The p=1 Gagliardo-Nirenberg-Sobolev inequality

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 2$ . There is a constant $C (n)$ such that $∥ u ∥_{L^{n / (n - 1)} (R^{n})} \leq C (n) ∥ D u ∥_{L^{1} (R^{n})}$ for every $u \in C_{c}^{\infty} (R^{n}; K)$ .

Facts & Assumptions

Given: Countable Choice; an integer $n \geq 2$ ; a field $K \in {R, C}$ ; and a function $u \in C_{c}^{\infty} (R^{n}; K)$ .

[F1]

Vector-valued fundamental theorem: if $f$ is differentiable with integrable derivative on an interval, then $\int_{a}^{b} f^{'} = f (b) - f (a)$ (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz).

[F2]

Tonelli's theorem on sigma-finite products: iterated integrals of nonnegative product-measurable functions may be computed in any order and partial integrals may be renamed (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F3]

Holder's inequality: if $1 / r = 1 / p + 1 / q$ and the indicated spaces are over one measure space, then $∥ f g ∥_{r} \leq ∥ f ∥_{p} ∥ g ∥_{q}$ (Generalized Holder inequality puts products into $L^{r}$ ).

[F4]

$L^{p}$ is the quotient by almost-everywhere null functions, and complex-valued Lebesgue spaces use the componentwise conventions (The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions).

[F5]

Countable Choice, assumed for the measure-theoretic interfaces above (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1F1F4givenalgebra

Sections and pointwise bounds. Fix $j \in {1, \dots, n}$ and write ${\hat{x}}_{j}$ for the coordinates other than $x_{j}$ . For each fixed ${\hat{x}}_{j}$ , the section $t \mapsto u (x_{1}, \dots, t, \dots, x_{n})$ is smooth and compactly supported, so the fundamental theorem [F1] applied on an interval containing the support and the estimate $∣ \partial_{j} u ∣ \leq ∣ D u ∣$ give $∣ u (x) ∣ \leq \int_{R} ∣ D u (x_{1}, \dots, t, \dots, x_{n}) ∣ d t = : P_{j} ({\hat{x}}_{j})$ for every $x$ .

1.2F2F3algebra

The product-integral lemma. For $d \geq 2$ and nonnegative integrable functions $f_{j}$ on $R^{d - 1}$ , each independent of the $j$ -th coordinate, one has $\int_{R^{d}} \prod_{j = 1}^{d} f_{j} ({\hat{x}}_{j})^{1 / (d - 1)} d x \leq \prod_{j = 1}^{d} (\int f_{j})^{1 / (d - 1)}$ . For $d = 2$ this is Tonelli [F2]. For $d > 2$ , integrate first in $x_{d}$ and apply [F3] with $d - 1$ equal exponents to the factors $j < d$ . Put $g_{j} : = \int_{R} f_{j} d x_{d}$ for $j < d$ ; the resulting upper bound is $\int_{R^{d - 1}} f_{d}^{1 / (d - 1)} \prod_{j < d} g_{j}^{1 / (d - 1)}$ . Holder with exponents $d - 1$ and $(d - 1) / (d - 2)$ bounds this by $(\int f_{d})^{1 / (d - 1)} (\int \prod_{j < d} g_{j}^{1 / (d - 2)})^{(d - 2) / (d - 1)}$ . The induction hypothesis in dimension $d - 1$ , followed by Tonelli, gives the required product of the $\int f_{j}$ . Zero integrals make the integrand zero almost everywhere, so they cause no division.

2.1F5step 1.1algebra

Product and root. Multiplying the $n$ pointwise inequalities of step 1.1 and taking the $(n - 1)$ -th root gives $∣ u (x) ∣^{n / (n - 1)} \leq \prod_{j = 1}^{n} P_{j} ({\hat{x}}_{j})^{1 / (n - 1)}$ for every $x$ .

3.1F2F4step 2.1step 1.2algebra∎

Apply step 1.2 with $d = n$ and $f_{j} = P_{j}$ . Tonelli gives $\int_{R^{n - 1}} P_{j} = \int_{R^{n}} ∣ D u ∣ = ∥ D u ∥_{1} < \infty$ . Step 2.1 therefore yields $\int ∣ u ∣^{n / (n - 1)} \leq ∥ D u ∥_{1}^{n / (n - 1)}$ . Taking the $(n - 1) / n$ -th power proves the assertion with $C (n) = 1$ .

Source notes

Kinnunen's Theorem 3.3 computes the product of the $n$ one-dimensional primitive estimates and integrates one variable at a time with the generalized Holder inequality for $(n - 1)$ factors; the proof above records a dimension induction for the product-integral inequality. The constant obtained is $1$ , which is not sharp but is dimension-only as asserted. The argument is the case $p = 1$ separated in the plan because the power-and-Holder reduction used for $1 < p < n$ is unavailable at the endpoint.

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The Gagliardo-Nirenberg-Sobolev inequality for $1 < p < n$

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $n \geq 2$ , $1 < p < n$ and $p^{*} = \frac{n p}{n - p}$ . There is a constant $C (n, p)$ such that $∥ u ∥_{L^{p^{*}} (R^{n})} \leq C (n, p) ∥ D u ∥_{L^{p} (R^{n})}$ for every $u \in W^{1, p} (R^{n}; K)$ ; here $∣ D u ∣$ is the Euclidean norm of the weak gradient.

Facts & Assumptions

Given: The Axiom of Choice, whose Countable-Choice consequence is used for the density and completeness interfaces; integers $n \geq 2$ and an exponent $1 < p < n$ ; the conjugate $p^{*} = n p / (n - p)$ ; a field $K \in {R, C}$ .

[F1]

The Sobolev conjugate satisfies $p^{*} > p$ and $\frac{1}{p^{*}} = \frac{1}{p} - \frac{1}{n}$ , so $γ : = \frac{p (n - 1)}{n - p} > 1$ and $(γ - 1) \frac{p}{p - 1} = p^{*}$ , $γ \frac{n}{n - 1} = p^{*}$ (The Sobolev conjugate exponent and the scaling identity).

[F2]

The endpoint inequality: there is $C_{1} (n)$ with $∥ v ∥_{L^{n / (n - 1)} (R^{n})} \leq C_{1} (n) ∥ D v ∥_{L^{1} (R^{n})}$ for every $v \in C_{c}^{\infty} (R^{n}; K)$ (The p=1 Gagliardo-Nirenberg-Sobolev inequality).

[F3]

Holder's inequality in the form $\int ∣ f g ∣ \leq ∥ f ∥_{r} ∥ g ∥_{s}$ for conjugate exponents $r, s$ (Holder's inequality for integrals, including the endpoint cases).

[F4]

$C_{c}^{\infty} (R^{n}; K)$ is dense in $W^{1, p} (R^{n}; K)$ for $1 \leq p < \infty$ , whose elements are $L^{p}$ classes with weak gradients in $L^{p}$ (Compactly supported smooth functions are dense in W^{k,p}(R^n), Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions).

[F5]

$L^{p^{*}}$ is complete and every norm-convergent sequence has an almost-everywhere convergent subsequence (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , Complex Lp completeness and almost-everywhere subsequences).

[F6]

The classical chain rule computes the gradient of a smooth composition, and for smooth functions the classical derivatives are the weak derivatives (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Classical derivatives agree with weak derivatives).

[F7]

Dominated convergence: pointwise almost-everywhere convergence under one integrable majorant implies convergence in $L^{1}$ (Dominated convergence).

[F8]

A compact subset of an open set admits a smooth cutoff equal to $1$ on it (A Euclidean bump for a compact set inside an open set). Apply this to a compact neighbourhood of $supp u$ inside a bounded open ball. The resulting support is closed and bounded, hence compact, and the cutoff equals $1$ near $supp u$ .

Proof

1.1F1F2F6F8choosealgebra

The smooth real case. Let $u \in C_{c}^{\infty} (R^{n}; R)$ and choose $χ \in C_{c}^{\infty} (R^{n})$ with $0 \leq χ \leq 1$ and $χ = 1$ on a neighborhood of $supp u$ . Put $s : = ∣ u ∣$ , $q : = n / (n - 1)$ , and $v_{δ} : = χ (u^{2} + δ^{2})^{γ / 2}$ for $0 < δ \leq 1$ . Then $v_{δ} \in C_{c}^{\infty}$ . By [F6], on $supp u$ its gradient has modulus $M_{δ} : = γ s (s^{2} + δ^{2})^{(γ - 2) / 2} ∣ D u ∣$ , while off that support the only derivative term is $δ^{γ} D χ$ . Thus [F2] gives $∥ v_{δ} ∥_{L^{q}} \leq C_{1} (n) (\int M_{δ} + δ^{γ} \int ∣ D χ ∣)$ .

1.2F1F2F6F7algebra

Let $δ_{j} ↓ 0$ . Since $v_{δ_{j}} \to ∣ u ∣^{γ}$ pointwise and is dominated by the bounded compactly supported function $χ (s^{2} + 1)^{γ / 2}$ , [F7] gives $∥ v_{δ_{j}} ∥_{L^{q}} \to ∥ u ∥_{L^{p^{*}}}^{γ}$ . Also $M_{δ_{j}} \to γ s^{γ - 1} ∣ D u ∣$ almost everywhere. If $1 < γ \leq 2$ , then $M_{δ} \leq γ s^{γ - 1} ∣ D u ∣$ ; if $γ > 2$ , then for $0 < δ \leq 1$ , $M_{δ} \leq C_{γ} (s^{γ - 1} + s) ∣ D u ∣$ . These majorants are integrable because $u$ is smooth and compactly supported, so [F7] gives $\int M_{δ_{j}} \to γ \int s^{γ - 1} ∣ D u ∣$ , while $δ_{j}^{γ} \int ∣ D χ ∣ \to 0$ . Taking limits in the endpoint estimate yields $∥ u ∥_{L^{p^{*}}}^{γ} \leq C_{1} (n) γ \int s^{γ - 1} ∣ D u ∣$ .

2.1step 1.2F1F3algebra

Holder [F3] and $(γ - 1) p / (p - 1) = p^{*}$ [F1] give $\int s^{γ - 1} ∣ D u ∣ \leq ∥ u ∥_{L^{p^{*}}}^{γ - 1} ∥ D u ∥_{L^{p}}$ . If $u \neq 0$ in $L^{p^{*}}$ , divide by $∥ u ∥_{L^{p^{*}}}^{γ - 1}$ ; if $u = 0$ almost everywhere, the estimate is immediate. Thus the real smooth case holds with constant $C_{1} (n) γ$ .

3.1step 2.1algebra

The smooth complex case. Let $u \in C_{c}^{\infty} (R^{n}; C)$ with real and imaginary parts $a$ and $b$ . Applying step 2.1 to both parts and using $∣ u ∣ \leq ∣ a ∣ + ∣ b ∣$ gives $∥ u ∥_{L^{p^{*}}} \leq C_{1} (n) γ (∥ D a ∥_{L^{p}} + ∥ D b ∥_{L^{p}}) \leq 2 C_{1} (n) γ ∥ D u ∥_{L^{p}}$ .

4.1F4F5step 2.1step 3.1algebra∎

Passage to $W^{1, p}$ by density. Let $u \in W^{1, p} (R^{n}; K)$ . By [F4] choose $u_{k} \in C_{c}^{\infty} (R^{n}; K)$ with $u_{k} \to u$ in $W^{1, p} (R^{n})$ . Applying step 2.1 (real case) or step 3.1 (complex case) to the differences gives $∥ u_{k} - u_{j} ∥_{L^{p^{*}}} \leq C^{'} (n, p) ∥ D (u_{k} - u_{j}) ∥_{L^{p}}$ , so $(u_{k})$ is Cauchy in $L^{p^{*}}$ . By [F5] it converges in $L^{p^{*}}$ to a class $w$ , and a subsequence converges to $w$ almost everywhere. Since $u_{k} \to u$ in $L^{p}$ , a further subsequence converges to $u$ almost everywhere, so $w = u$ almost everywhere. Passing to the limit in the smooth inequality gives $∥ u ∥_{L^{p^{*}}} \leq C^{'} (n, p) ∥ D u ∥_{L^{p}}$ , and renaming this constant proves the claim.

Source notes

This is Kinnunen's Theorem 3.3 for $1 < p < n$ , printed pp. 63-65: the device is to apply the endpoint ( $p = 1$ ) inequality to $v = ∣ u ∣^{γ}$ with $γ = p (n - 1) / (n - p)$ and to use the Holder pairing $(γ - 1) p / (p - 1) = p^{*}$ . The smooth compact cutoff $χ (u^{2} + δ^{2})^{γ / 2}$ makes the endpoint application legitimate; dominated convergence removes the regularisation, including the cutoff-gradient term, before the density passage. Hunter's Theorems 3.28 and 3.31 and Teschl's Theorem 9.22 record the same proof; Laugesen's Theorem 3.17 is the endpoint form used here as the p=1 input.

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The Sobolev inequality for zero-boundary Sobolev closures on open sets

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $Ω \subseteq R^{n}$ be open, $n \geq 2$ , $1 \leq p < n$ and $p^{*} = \frac{n p}{n - p}$ . There is $C (n, p)$ with $∥ u ∥_{L^{p^{*}} (Ω)} \leq C (n, p) ∥ D u ∥_{L^{p} (Ω)}$ for every $u \in W_{0}^{1, p} (Ω; K)$ .

Facts & Assumptions

Given: The Axiom of Choice; an open set $Ω \subseteq R^{n}$ ; $n \geq 2$ ; $1 \leq p < n$ ; a field $K \in {R, C}$ ; and a class $u \in W_{0}^{1, p} (Ω; K)$ .

[F1]

$W_{0}^{1, p} (Ω)$ is the closure of $C_{c}^{\infty} (Ω)$ in the $W^{1, p}$ norm, and its elements are $L^{p}$ classes with weak gradients in $L^{p}$ (Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms).

[F2]

Extension by zero sends $W_{0}^{1, p} (Ω; K)$ into $W^{1, p} (R^{n}; K)$ , the weak derivatives of the extension are the zero extensions of the weak derivatives, and all $L^{p}$ component norms are preserved (Zero extension of W_0^{1,p} has no boundary derivative).

[F3]

For $1 < p < n$ , the whole-space inequality holds for every $v \in W^{1, p} (R^{n})$ (The Gagliardo-Nirenberg-Sobolev inequality for $1 < p < n$ ). For $p = 1$ it holds for $v \in C_{c}^{\infty}$ (The p=1 Gagliardo-Nirenberg-Sobolev inequality). Here $p^{*} = n p / (n - p)$ (The Sobolev conjugate exponent and the scaling identity).

[F4]

Every $L^{r}$ space for $1 \leq r \leq \infty$ is complete and norm convergence has an almost-everywhere convergent subsequence (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , Complex Lp completeness and almost-everywhere subsequences).

Proof

technique · direct

1.1F1F2givenalgebra

Zero extension. Let $E_{0} u$ be the extension of $u$ by zero. By [F2], $E_{0} u \in W^{1, p} (R^{n}; K)$ , its weak gradient is the zero extension of $D u$ , and $∥ E_{0} u ∥_{L^{p^{*}} (R^{n})} = ∥ u ∥_{L^{p^{*}} (Ω)}$ , $∥ D (E_{0} u) ∥_{L^{p} (R^{n})} = ∥ D u ∥_{L^{p} (Ω)}$ componentwise.

2.1F1F2F3F4step 1.1algebra∎

For $1 < p < n$ , apply [F3] to $E_{0} u$ and use step 1.1. For $p = 1$ , choose $φ_{j} \in C_{c}^{\infty} (Ω)$ converging to $u$ in $W^{1, 1} (Ω)$ by [F1]; their zero extensions converge to $E_{0} u$ in $W^{1, 1} (R^{n})$ by [F2]. The endpoint estimate [F3] applied to differences shows that these extensions are Cauchy in $L^{n / (n - 1)}$ . By [F4] their limit in that space exists; an almost-everywhere subsequence, followed by an $L^{1}$ almost-everywhere subsequence, identifies it with $E_{0} u$ . Passing to the limit in the endpoint estimate gives $∥ E_{0} u ∥_{n / (n - 1)} \leq C (n) ∥ D (E_{0} u) ∥_{1}$ . Step 1.1 transfers both cases to $Ω$ , proving the assertion.

Source notes

The corollary is the zero-trace case of the whole-space Sobolev inequality, Kinnunen's Remark 3.4(3) and Laugesen's Theorem 3.18: the extension by zero has the same weak gradient up to the boundary of $Ω$ , so the whole-space result transfers verbatim. At $p = 1$ the smooth endpoint estimate is extended by the closure approximation and $L^{n / (n - 1)}$ completeness.

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Poincare inequality on a ball

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $n \geq 1$ , $B = B (x_{0}, r) \subseteq R^{n}$ with $r > 0$ , $1 \leq p < \infty$ , and $u \in W^{1, p} (B; K)$ with ball average $u_{B} = ∣ B ∣^{- 1} \int_{B} u$ . Then $∥ u - u_{B} ∥_{L^{p} (B)} \leq C (n, p) r ∥ D u ∥_{L^{p} (B)}$ .

Facts & Assumptions

Given: The Axiom of Choice; an integer $n \geq 1$ ; a ball $B = B (x_{0}, r) \subseteq R^{n}$ with $r > 0$ ; an exponent $1 \leq p < \infty$ ; a field $K \in {R, C}$ ; and a class $u \in W^{1, p} (B; K)$ .

[F1]

The convex-domain Poincare-Wirtinger estimate: for every open bounded convex nonempty $Ω$ and every $v \in W^{1, p} (Ω; K)$ , $∥ v - v_{Ω} ∥_{L^{p} (Ω)} \leq C (n) diam (Ω) ∥ D v ∥_{L^{p} (Ω)}$ with $C (n) = 2 (2^{n} - 1) / n$ , where $v_{Ω} = ∣ Ω ∣^{- 1} \int_{Ω} v$ (Poincare-Wirtinger on bounded convex domains by the direct pairwise argument).

[F2]

The ball average $u_{B} = ∣ B ∣^{- 1} \int_{B} u$ is the mean of $u$ over $B$ ; it is defined because every ball has positive finite Lebesgue measure (The average of a locally integrable function over a Euclidean ball, Euclidean balls have positive finite Lebesgue measure).

[F3]

$W^{1, p} (B; K)$ consists of the $L^{p}$ classes with weak first derivatives in $L^{p}$ , and $L^{p}$ consists of almost-everywhere classes (Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions).

Proof

technique · direct

1.1F2F3givenalgebra

The ball is admissible for [F1]. The ball $B = B (x_{0}, r)$ is open, bounded, convex and nonempty, and its diameter is $diam (B) = 2 r$ ; the class $u$ lies in $W^{1, p} (B; K)$ by hypothesis. Its mean over $B$ as a convex set is exactly the ball average $u_{B} = ∣ B ∣^{- 1} \int_{B} u$ of [F2], because both are $∣ B ∣^{- 1} \int_{B} u$ ; the value is a finite element of $K$ by [F2] and [F3].

2.1F1step 1.1algebra∎

Applying the convex-domain estimate. By [F1] applied to $Ω = B$ and $v = u$ , $∥ u - u_{B} ∥_{L^{p} (B)} \leq C (n) diam (B) ∥ D u ∥_{L^{p} (B)} = 2 C (n) r ∥ D u ∥_{L^{p} (B)}$ . Hence the asserted inequality holds with the dimension-and-exponent constant $C (n, p) : = 2 C (n) = 4 (2^{n} - 1) / n$ , which depends only on $n$ and $p$ .

Source notes

Kinnunen proves the ball case by the pointwise potential estimate and the maximal-function bound; Laugesen records it as an exercise with a constant linear in $r$ . The proof above derives the ball statement from the more general convex-domain Poincare-Wirtinger corollary proved earlier on this page, with the explicit constant $4 (2^{n} - 1) / n$ , which is not sharp but is dimension-only and linear in $r$ as asserted.

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The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $Ω \subseteq R^{n}$ be open and bounded in one direction: there are a unit vector $e$ and $a < b$ with $a < x \cdot e < b$ for every $x \in Ω$ . Let $1 \leq p < \infty$ . Then $∥ u ∥_{L^{p} (Ω)} \leq C (p) (b - a) ∥ D u ∥_{L^{p} (Ω)}$ for every $u \in W_{0}^{1, p} (Ω; K)$ .

Facts & Assumptions

Given: Countable Choice; an open set $Ω \subseteq R^{n}$ bounded in the direction $e \in S^{n - 1}$ between $a < b$ ; an exponent $1 \leq p < \infty$ ; a field $K \in {R, C}$ ; and a class $u \in W_{0}^{1, p} (Ω; K)$ .

[F1]

$W_{0}^{1, p} (Ω; K)$ is the closure in $W^{1, p} (Ω; K)$ of the compactly supported smooth functions $C_{c}^{\infty} (Ω; K)$ , and every such smooth function lies in $W^{1, p}$ with its classical derivatives as weak derivatives (Zero-boundary Sobolev space as a norm closure).

[F2]

Vector-valued fundamental theorem: if $f : [a, b] \to R^{m}$ is differentiable with integrable derivative, then $\int_{a}^{b} f^{'} = f (b) - f (a)$ (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz).

[F3]

Holder's inequality for integrals (Holder's inequality for integrals, including the endpoint cases).

[F4]

Tonelli's theorem on sigma-finite products, allowing iterated integrals of nonnegative measurable functions in either order (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F5]

Linear change of variables: an invertible linear map $T$ of $R^{n}$ scales Lebesgue measure by $∣ \det T ∣$ and the integral substitution formula holds for nonnegative Borel integrands; in particular an orthogonal change of orthonormal coordinates preserves the integral (A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not).

[F6]

$W^{1, p}$ consists of $L^{p}$ classes with weak derivatives in $L^{p}$ and the norm is the $ℓ^{p}$ norm of $u$ and its coordinate weak derivatives (Integer-order Sobolev spaces and their norms); $L^{p}$ classes are almost-everywhere classes (The space $L^{p} (μ)$ as the quotient by null functions).

[F7]

Countable Choice, assumed for the measure and closure interfaces above (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1F2F3givenalgebra

The smooth case: pointwise bound. Let $φ \in C_{c}^{\infty} (Ω; K)$ and extend it by zero to $R^{n}$ . Write $x = x_{⊥} + x_{n} e$ with $x_{⊥} ⊥ e$ . For each fixed $x_{⊥}$ , the profile $g (s) : = φ (x_{⊥} + s e)$ is smooth and supported in $(a, b)$ , since $Ω \subset {a < y \cdot e < b}$ . If $x = x_{⊥} + x_{n} e \in Ω$ , then $a < x_{n} < b$ and $g (a) = 0$ . Applying [F2] componentwise (in $R$ or $R^{2}$ according to the scalar field) gives $φ (x) = g (x_{n}) - g (a) = \int_{a}^{x_{n}} D φ (x_{⊥} + s e) \cdot e d s$ . Hence $∣ φ (x) ∣ \leq \int_{a}^{x_{n}} ∣ D φ (x_{⊥} + s e) ∣ d s$ . Holder [F3], followed by $x_{n} - a \leq b - a$ and enlargement of the integration interval, yields $∣ φ (x) ∣^{p} \leq (b - a)^{p - 1} \int_{a}^{b} ∣ D φ (x_{⊥} + s e) ∣^{p} d s$ .

2.1F4F5step 1.1algebra

The smooth case: integration. If $n = 1$ , then $∣ Ω ∣ \leq b - a$ , so integrating step 1.1 gives $\int_{Ω} ∣ φ ∣^{p} \leq (b - a)^{p} \int_{a}^{b} ∣ D φ (s e) ∣^{p} d s = (b - a)^{p} \int_{Ω} ∣ D φ ∣^{p}$ by the change of variable $y = s e$ and the support of $D φ$ in $Ω$ . Now suppose $n \geq 2$ . Choose orthonormal coordinates with last vector $e$ and write $x = x_{⊥} + x_{n} e$ ; by [F5] integration on $R^{n}$ is integration over $(x_{⊥}, x_{n}) \in R^{n - 1} \times R$ . For each $x_{⊥}$ , the slice $S_{x_{⊥}} : = {x_{n} : (x_{⊥}, x_{n}) \in Ω}$ is a measurable subset of $(a, b)$ , so $∣ S_{x_{⊥}} ∣ \leq b - a$ . Integrating step 1.1 over $Ω$ and applying Tonelli [F4] gives $\int_{Ω} ∣ φ ∣^{p} d x \leq (b - a)^{p - 1} \int_{R^{n - 1}} \int_{S_{x_{⊥}}} \int_{a}^{b} ∣ D φ (x_{⊥} + s e) ∣^{p} d s d x_{n} d x_{⊥} \leq (b - a)^{p} \int_{R^{n - 1}} \int_{a}^{b} ∣ D φ (x_{⊥} + s e) ∣^{p} d s d x_{⊥}$ . The last integral equals $\int_{R^{n}} ∣ D φ ∣^{p} d x = \int_{Ω} ∣ D φ ∣^{p}$ , since $φ$ and its gradient vanish outside $Ω$ and $Ω$ lies in the slab. Therefore $∥ φ ∥_{L^{p} (Ω)}^{p} \leq (b - a)^{p} ∥ D φ ∥_{L^{p} (Ω)}^{p}$ .

3.1F1F6F7step 2.1algebra∎

The general class. By [F1] there are $φ_{k} \in C_{c}^{\infty} (Ω; K)$ with $φ_{k} \to u$ in $W^{1, p} (Ω; K)$ ; by step 2.1, $∥ φ_{k} ∥_{L^{p}} \leq (b - a) ∥ D φ_{k} ∥_{L^{p}}$ for every $k$ . Both sides are continuous in the $W^{1, p}$ norm: $∥ φ_{k} ∥_{p} \to ∥ u ∥_{p}$ and $∥ D φ_{k} ∥_{p} \to ∥ D u ∥_{p}$ by [F6]. Passing to the limit gives $∥ u ∥_{L^{p} (Ω)} \leq (b - a) ∥ D u ∥_{L^{p} (Ω)}$ , so the asserted inequality holds with $C (p) = 1$ .

Source notes

Kinnunen's Theorem 3.10 proves the estimate on bounded open sets by taking the primitive in one coordinate direction and applying Holder; the proof above runs the same argument along the unit vector $e$ of the hypothesis, uses the orthonormal coordinate decomposition for the integration, and then extends from $C_{c}^{\infty} (Ω)$ to $W_{0}^{1, p} (Ω)$ by the definition of the latter as a closure. The constant obtained is $1$ , independent of $p$ ; the statement permits a $p$ -dependent constant.

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Mean-zero Poincare estimate on bounded connected extension domains below the dimension

Statement

Assume the Axiom of Choice (and hence Countable Choice and Dependent Choice). Let $n \geq 2$ , $1 < p < n$ , $K \in {R, C}$ , and let $Ω \subset R^{n}$ be a nonempty bounded connected $W^{1, p}$ -extension domain. Then there exists $C_{P} = C_{P} (n, p, Ω, K)$ such that $∥ u - u_{Ω} ∥_{L^{p} (Ω)} \leq C_{P} ∥ D u ∥_{L^{p} (Ω)}$ for every $u \in W^{1, p} (Ω; K)$ , where $u_{Ω} = ∣ Ω ∣^{- 1} \int_{Ω} u$ .

Facts & Assumptions

Given: The Axiom of Choice; integers $n \geq 2$ ; an exponent $1 < p < n$ ; a field $K \in {R, C}$ ; a nonempty bounded connected $W^{1, p}$ -extension domain $Ω \subset R^{n}$ ; a bounded linear extension operator $E : W^{1, p} (Ω; K) \to W^{1, p} (R^{n}; K)$ with $(E u) ∣_{Ω} = u$ and operator norm $∥ E ∥$ ; a nonnegative unit-mass $ρ \in C_{c}^{\infty} (R^{n})$ with $supp ρ \subseteq {\overline{B}}_{1} (0)$ and radial mollifiers $ρ_{ε}$ .

[F1]

The Axiom of Choice is the statement that every family of nonempty sets has a choice function, and it implies Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F2]

Dependent Choice is the statement that every entire relation on a nonempty set admits a sequence with prescribed first term (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F3]

$W^{1, p} (Ω; K)$ consists of the $L^{p}$ classes with weak first derivatives in $L^{p}$ , and $L^{p}$ is the quotient by almost-everywhere null functions (Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions).

[F4]

A $W^{1, p}$ -extension domain carries a bounded linear $E$ with $(E u) ∣_{Ω} = u$ (Sobolev extension domains and extension operators).

[F5]

Weak Leibniz rule: for smooth $η$ with bounded value and first derivatives and $v \in W^{1, p}$ the product $η v$ lies in $W^{1, p}$ and $D (η v) = (D η) v + η D v$ (Weak Leibniz rule with a smooth factor).

[F6]

For a compact $K$ inside an open $U$ there is a smooth $χ$ with $χ = 1$ on $K$ and support in $U$ (A Euclidean bump for a compact set inside an open set).

[F7]

$C_{c}^{\infty} (R^{n}; K)$ is dense in $W^{1, p} (R^{n}; K)$ for $1 \leq p < \infty$ (Compactly supported smooth functions are dense in W^{k,p}(R^n)).

[F8]

Lebesgue measure is translation invariant (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).

[F9]

Vector-valued fundamental theorem: for a differentiable $f : [a, b] \to R^{m}$ with integrable derivative, $f (b) - f (a) = \int_{a}^{b} f^{'}$ (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz).

[F10]

The family $ρ_{ε} (x) = ε^{- n} ρ (x / ε)$ is the radial mollifier family generated by $ρ$ , with $ρ_{ε} \geq 0$ , $\int ρ_{ε} = 1$ and $supp ρ_{ε} \subseteq {\overline{B}}_{ε} (0)$ (A radial mollifier family in Rn).

[F11]

For locally integrable $f$ the convolution $f * ρ_{ε}$ is smooth with $\partial^{α} (f * ρ_{ε}) = f * (\partial^{α} ρ_{ε})$ (Convolution with a mollifier is smooth, and derivatives pass under the integral sign).

[F12]

Holder's inequality for integrals (Holder's inequality for integrals, including the endpoint cases).

[F13]

Minkowski's integral inequality: for a measurable $F$ on a product with $\int_{Y} ∥ F (\cdot, y) ∥_{L^{p} (X)} d ν (y) < \infty$ , the function $x \mapsto \int_{Y} ∣ F (x, y) ∣ d ν (y)$ lies in $L^{p} (X)$ with norm at most $\int_{Y} ∥ F (\cdot, y) ∥_{L^{p} (X)} d ν (y)$ (Minkowski's integral inequality).

[F14]

Arzela-Ascoli: for a nonempty compact metric space $K$ , a subset of $C (K, R)$ has compact closure in the supremum metric exactly when it is equicontinuous and pointwise bounded (Arzelà--Ascoli for real $C (K)$ under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded).

[F15]

Under Countable Choice and Dependent Choice a compact metric space is sequentially compact, and Euclidean closed balls are compact (For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice, For $n \geq 1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact).

[F16]

$L^{p}$ is complete for $1 \leq p \leq \infty$ (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , Complex Lp completeness and almost-everywhere subsequences).

[F17]

Weak derivatives are unique almost everywhere, and the weak derivative is defined by the test-function identity (Uniqueness of a weak derivative as an almost-everywhere class, Weak derivative of a locally integrable function).

[F18]

If $u \in W_{l o c}^{1, p} (Ω; K)$ has $D_{i} u = 0$ almost everywhere for every $i$ , then $u$ is almost everywhere constant on each connected component of $Ω$ (Zero weak gradient gives componentwise constants).

[F19]

On a completed sigma-finite product, nonnegative measurable functions may be integrated in either order, and Tonelli-Fubini applies to measurable integrands of the form $G (x, y)$ (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability).

[F20]

Every Euclidean ball has positive finite Lebesgue measure (Euclidean balls have positive finite Lebesgue measure), $λ$ is monotone under inclusion (Measures are monotone), and bounded subsets have finite outer measure (Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure).

[F21]

Under Countable Choice, Lebesgue measure is the completion of its Borel restriction ( $L (R^{n})$ is exactly the completion of the restriction of $λ_{n}$ to the Borel sets), and every completion-measurable real function has an almost-everywhere equal Borel representative (A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra); apply this to real and imaginary parts for complex functions.

Proof

technique · contradiction

1.1F1F3F20givenchoose

The contradiction setup. Because $Ω$ is nonempty open and bounded, it contains a ball and $0 < ∣ Ω ∣ < \infty$ by [F20], so the mean $u_{Ω}$ is defined for every $u \in L^{p} (Ω; K)$ . Suppose the asserted constant does not exist. Then for every $j \geq 1$ there is $u_{j} \in W^{1, p} (Ω; K)$ with $∥ u_{j} - (u_{j})_{Ω} ∥_{L^{p} (Ω)} > j ∥ D u_{j} ∥_{L^{p} (Ω)}$ ; the left side is positive, and Countable Choice [F1] selects such a sequence. Put $v_{j} : = (u_{j} - (u_{j})_{Ω}) / ∥ u_{j} - (u_{j})_{Ω} ∥_{L^{p} (Ω)}$ . Then $v_{j} \in W^{1, p} (Ω; K)$ , $(v_{j})_{Ω} = 0$ , $∥ v_{j} ∥_{L^{p} (Ω)} = 1$ and $∥ D v_{j} ∥_{L^{p} (Ω)} < 1 / j$ .

1.2F7F8F9F13algebra

Translation differences. Every $W \in W^{1, p} (R^{n}; K)$ and $h \in R^{n}$ satisfy $∥ W (\cdot - h) - W ∥_{L^{p} (R^{n})} \leq ∣ h ∣ ∥ D W ∥_{L^{p} (R^{n})}$ . For smooth compactly supported $φ$ the fundamental theorem [F9] applied to $t \mapsto φ (x - t h)$ gives $φ (x - h) - φ (x) = - \int_{0}^{1} D φ (x - t h) h d t$ , so Minkowski [F13] and translation invariance [F8] give $∥ φ (\cdot - h) - φ ∥_{p} \leq ∣ h ∣ \int_{0}^{1} ∥ D φ (\cdot - t h) ∥_{p} d t = ∣ h ∣ ∥ D φ ∥_{p}$ . For general $W$ , [F7] provides $φ_{k} \in C_{c}^{\infty}$ with $φ_{k} \to W$ in $W^{1, p} (R^{n})$ ; applying the smooth bound to $φ_{k}$ and letting $k \to \infty$ , using translation invariance [F8] on the left and strong convergence on the right, gives the claim.

2.1F4F5F6step 1.1algebra

Extension and cutoff. Fix $E$ as in [F4] and put $M_{0} : = ∥ E ∥$ . The closure $\overline{Ω}$ is compact. Choose a bounded open ball $U$ containing it; [F6] gives a smooth $χ$ equal to $1$ on $\overline{Ω}$ with closed support contained in $U$ . That support is bounded and hence compact, so $χ \in C_{c}^{\infty} (R^{n})$ ; put $K : = supp χ$ , a compact set. Define $F_{j} : = χ E v_{j}$ . By [F5] each $F_{j}$ lies in $W^{1, p} (R^{n}; K)$ , has support in $K$ , restricts to $v_{j}$ on $Ω$ (because $χ = 1$ there) and satisfies $∥ F_{j} ∥_{L^{p} (R^{n})} + ∥ D F_{j} ∥_{L^{p} (R^{n})} \leq (n + 1) (1 + ∥ χ ∥_{\infty} + ∥ D χ ∥_{\infty}) ∥ E v_{j} ∥_{W^{1, p} (R^{n})} \leq M$ with $M : = (n + 1)^{2} (1 + ∥ χ ∥_{\infty} + ∥ D χ ∥_{\infty}) M_{0}$ , using $∥ v_{j} ∥_{W^{1, p}} \leq n + 1$ and the elementary bound of the Euclidean gradient norm by the sum of its coordinate norms from step 1.1 and the operator bound of [F4].

3.1F10F11F12step 2.1algebra

Mollification and equicontinuity at one scale. Fix $0 < ε \leq 1$ . By [F10] and [F11], $F_{j} * ρ_{ε}$ is smooth on $R^{n}$ with $D (F_{j} * ρ_{ε}) = F_{j} * D ρ_{ε}$ and support in the compact set $K^{'} : = K + {\overline{B}}_{1} (0)$ . Holder [F12] gives, uniformly in $j$ and $x$ , $∣ (F_{j} * ρ_{ε}) (x) ∣ \leq ∥ F_{j} ∥_{L^{p}} ∥ ρ_{ε} ∥_{L^{p^{'}}} \leq M ∥ ρ_{ε} ∥_{L^{p^{'}}}$ and $∣ D (F_{j} * ρ_{ε}) (x) ∣ \leq M ∥ D ρ_{ε} ∥_{L^{p^{'}}}$ , using step 2.1. The second bound makes the family ${F_{j} * ρ_{ε}}_{j}$ equicontinuous on the compact metric space $K^{'}$ and the first makes it pointwise bounded.

3.2F10F13F19F21step 1.2step 2.1algebra

Uniform mollification error. By [F10], $F_{j} * ρ_{ε} - F_{j} = \int ρ_{ε} (y) (F_{j} (\cdot - y) - F_{j}) d y$ on $R^{n}$ . Choose finite-valued Borel representatives of each $F_{j}$ using [F21], changing them to zero on a Borel null set and outside $K$ . Then $(x, y) \mapsto F_{j} (x - y)$ and $F_{j} (x)$ are Borel measurable, since subtraction and projection are continuous. The integrand is therefore product measurable, with measurable absolute section integrals by [F19]. Minkowski [F13], the translation bound of step 1.2 and $supp ρ_{ε} \subseteq {\overline{B}}_{ε} (0)$ give $∥ F_{j} * ρ_{ε} - F_{j} ∥_{L^{p} (R^{n})} \leq \int ρ_{ε} (y) ∥ F_{j} (\cdot - y) - F_{j} ∥_{L^{p}} d y \leq ε ∥ D F_{j} ∥_{L^{p}} \leq ε M$ , the last inequality by step 2.1.

4.1F14F15step 3.1given

One scale at a time. Fix $0 < ε \leq 1$ . The family ${F_{j} * ρ_{ε}}_{j}$ is uniformly bounded and equicontinuous on the compact set $K^{'}$ by step 3.1, so by Arzela-Ascoli [F14] its closure in $C (K^{'}; C) ≅ C (K^{'}; R)^{2}$ is compact; applying [F14] to the real and imaginary parts componentwise and then the sequential compactness of [F15], there is a subsequence $(j_{k})_{k}$ and $G_{ε} \in C (K^{'}; K)$ with $F_{j_{k}} * ρ_{ε} \to G_{ε}$ uniformly on $K^{'}$ .

5.1F2step 4.1chooseconstruct

Diagonalisation over the scales. Apply step 4.1 successively to the scales $ε_{m} = 2^{- m}$ , each time to the previously selected subsequence, and select the $m$ -th extracted subsequence at stage $m$ in such a way that the diagonal sequence $(j_{k})$ , where $j_{k}$ is the $k$ -th index of the $k$ -th subsequence, is strictly increasing; Dependent Choice [F2] formalises the recursion. Then for every fixed $m$ the tail $(F_{j_{k}} * ρ_{ε_{m}})_{k \geq m}$ is a subsequence of the $m$ -th extracted subsequence, hence converges uniformly on $K^{'}$ and in particular is Cauchy in $L^{p} (K^{'})$ .

6.1F16step 3.2step 5.1algebra

Cauchy and the $L^{p}$ limit. For $k, l \geq m$ , step 3.2 applied at scale $ε_{m}$ and the uniform convergence on $K^{'}$ of step 5.1 give $∥ F_{j_{k}} - F_{j_{l}} ∥_{L^{p} (R^{n})} \leq 2 M ε_{m} + ∥ F_{j_{k}} * ρ_{ε_{m}} - F_{j_{l}} * ρ_{ε_{m}} ∥_{L^{p} (K^{'})}$ . Given $δ > 0$ choose $m$ with $2 M 2^{- m} < δ / 2$ , then $k, l$ large enough that the second term is below $δ / 2$ ; hence $(F_{j_{k}})_{k}$ is Cauchy in $L^{p} (R^{n}; K)$ and converges by [F16] to some $F \in L^{p} (R^{n}; K)$ .

7.1F3F12step 1.1step 2.1step 6.1algebra

Properties of the limit. Restrict $v : = F ∣_{Ω}$ . Since $F_{j_{k}} ∣_{Ω} = v_{j_{k}}$ by step 2.1 and $∥ v - v_{j_{k}} ∥_{L^{p} (Ω)} \leq ∥ F - F_{j_{k}} ∥_{L^{p} (R^{n})} \to 0$ by step 6.1, the limit of the norms gives $∥ v ∥_{L^{p} (Ω)} = 1$ , and Holder [F12] with $∣ Ω ∣ < \infty$ gives $∣ \int_{Ω} v ∣ = \lim_{k} ∣ \int_{Ω} v_{j_{k}} ∣ = 0$ because every $v_{j_{k}}$ has mean zero (step 1.1). Thus $v \in L^{p} (Ω; K)$ has unit norm and mean zero.

8.1F3F12F17step 1.1step 7.1algebra

The weak gradient of the limit vanishes. Let $φ \in C_{c}^{\infty} (Ω)$ and $1 \leq i \leq n$ . Since $v_{j_{k}}$ is the weak $i$ -th derivative pair, $\int_{Ω} v_{j_{k}} \partial_{i} φ = - \int_{Ω} D_{i} v_{j_{k}} φ$ for every $k$ by [F17]. Holder [F12] gives $∣ \int_{Ω} D_{i} v_{j_{k}} φ ∣ \leq ∥ D_{i} v_{j_{k}} ∥_{L^{p}} ∥ φ ∥_{L^{p^{'}}} \leq j_{k}^{- 1} ∥ φ ∥_{L^{p^{'}}} \to 0$ , using step 1.1, and $\int_{Ω} v_{j_{k}} \partial_{i} φ \to \int_{Ω} v \partial_{i} φ$ by step 7.1 and Holder. Hence $\int_{Ω} v \partial_{i} φ = 0$ for every test function and every $i$ , so the zero function is a weak $i$ -th derivative of $v$ ; by uniqueness of weak derivatives [F17], $D_{i} v = 0$ almost everywhere on $Ω$ and $v \in W^{1, p} (Ω; K)$ .

9.1F18step 7.1step 8.1givencontradiction∎

The contradiction. By step 8.1 the class $v \in W^{1, p} (Ω; K)$ has all weak derivatives zero almost everywhere, so [F18] and the connectedness of $Ω$ give a constant $c \in K$ with $v = c$ almost everywhere on $Ω$ . Step 7.1 gives $0 = \int_{Ω} v = c ∣ Ω ∣$ , hence $c = 0$ and $v = 0$ almost everywhere, contradicting $∥ v ∥_{L^{p} (Ω)} = 1$ from step 7.1. Therefore a constant $C_{P} = C_{P} (n, p, Ω, K)$ with the asserted property exists.

Source notes

Kinnunen proves the mean-zero estimate as the inner step of Theorem 3.47 (printed pp. 90-91) using the Rellich-Kondrachov compactness theorem. The proof above replaces that compactness input by an internal argument: the extension operator of the definition of an extension domain, a fixed smooth cutoff, mollification at every scale, Arzela-Ascoli on a fixed compact set, a diagonal subsequence and the completeness of $L^{p}$ . The weak derivative of the limit is obtained from the test-function identity rather than from strong convergence of gradients, so no compactness theorem from the later compactness page is used. The argument uses Countable Choice to select the minimising sequence and Dependent Choice for the nested subsequences; both are supplied by the Axiom of Choice assumed in the Statement.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

John domains and the John constant

Definition

Assume Countable Choice. Let $n \geq 2$ and let $Ω \subseteq R^{n}$ be open, bounded and nonempty. Its boundary $\partial Ω$ is then a nonempty compact subset of $R^{n}$ contained in a sufficiently large closed ball (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space, For $n \geq 1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact), and the distance $dist (x, \partial Ω)$ is defined for every $x$ and is a $1$ -Lipschitz function of $x$ ( $∣ d (x, A) - d (y, A) ∣ \leq d (x, y)$ , so the distance to a fixed nonempty set is $1$ -Lipschitz).

Rectifiable curves, their arclength functions are used with the conventions of The arc-length function $s_{γ} (t) = L (γ ∣_{[a, t]})$ of a rectifiable path.

John domain and John constant. A pair $(Ω, x_{0})$ with $x_{0} \in Ω$ satisfies the John condition with constant $c \geq 1$ if for every $x \in Ω$ there is a rectifiable curve $γ : [0, l] \to Ω$ parametrised by arclength, with $γ (0) = x$ , $γ (l) = x_{0}$ and $dist (γ (t), \partial Ω) \geq c^{- 1} ∣ x - γ (t) ∣ for every t \in [0, l] .$ Write $c_{J} (Ω, x_{0})$ for the infimum of the admissible constants $c \geq 1$ ; with the convention $\inf \emptyset = + \infty$ , this value belongs to $[1, + \infty]$ . The domain $Ω$ is a John domain if $c_{J} (Ω, x_{0}) < \infty$ for some $x_{0} \in Ω$ , and $c_{J} (Ω, x_{0})$ is the John constant of the pair $(Ω, x_{0})$ . The estimates below use an admissible constant, never the finiteness of the infimum alone, and whether the infimum is attained is immaterial.

Arclength form. If $γ$ is arclength parametrised from $x$ , then $t \geq ∣ γ (t) - x ∣$ for $0 \leq t \leq l$ , because the straight segment from $x$ to $γ (t)$ is no longer than the curve. Hence a curve satisfying the arclength normalisation $dist (γ (t), \partial Ω) \geq c^{- 1} t$ for all $t$ also satisfies the displayed relative-distance condition with the same constant $c$ . Only this implication is used here; the chain construction below uses the displayed relative-distance condition.

Connectedness. A John domain is path-connected and hence connected: given $x, y \in Ω$ , choose curves $γ_{x}$ from $x$ to $x_{0}$ and $γ_{y}$ from $y$ to $x_{0}$ as in the definition (under Countable Choice the two curves may be chosen simultaneously) and traverse $γ_{x}$ followed by the reverse of $γ_{y}$ . No regularity of $\partial Ω$ is assumed beyond what the definition uses.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Bounded-overlap ball chains in a bounded John domain

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $n \geq 2$ and let $Ω \subseteq R^{n}$ be a bounded John domain with distinguished point $x_{0}$ and admissible constant $c_{J} \geq 1$ ; set $r_{0} = \frac{1}{4} dist (x_{0}, \partial Ω)$ and $B_{0} = B (x_{0}, r_{0})$ . There is a constant $M = M (n, c_{J}) \geq 1$ such that for every $x \in Ω$ there are balls $B_{i} = B (x_{i}, r_{i}) \subseteq Ω$ , $i \geq 0$ , with:

$∣ B_{i} \cup B_{i + 1} ∣ \leq M ∣ B_{i} \cap B_{i + 1} ∣$ for all $i \geq 0$ ;
$dist (x, B_{i}) \leq M r_{i}$ for all $i \geq 0$ , and $r_{i} \to 0$ , $x_{i} \to x$ as $i \to \infty$ ;
no point of $Ω$ belongs to more than $M$ of the balls $B_{i}$ .

For $x \notin B (x_{0}, 2 r_{0})$ the chain starts with $B_{0} = B (x_{0}, r_{0})$ and follows a John curve from $x_{0}$ to $x$ ; for $x \in B (x_{0}, 2 r_{0})$ it is the explicit geometric chain constructed below. The choice assumption supplies Countable Choice for the John-domain and Lebesgue-measure interfaces; selecting a curve for one fixed $x$ needs no choice axiom.

Facts & Assumptions

Given: The Axiom of Choice; $n \geq 2$ ; a bounded John domain $Ω$ with distinguished point $x_{0}$ and admissible constant $c_{J} \geq 1$ ; $r_{0} = \frac{1}{4} dist (x_{0}, \partial Ω) > 0$ ; and a point $x \in Ω$ .

[F1]

There is a path $γ : [0, 1] \to Ω$ with $γ (0) = x$ , $γ (1) = x_{0}$ and $dist (γ (t), \partial Ω) \geq c_{J}^{- 1} ∣ x - γ (t) ∣$ for every $t \in [0, 1]$ (John domains and the John constant). Evaluating at $t = 1$ gives $∣ x - x_{0} ∣ \leq c_{J} dist (x_{0}, \partial Ω) = 4 c_{J} r_{0}$ .

[F2]

For every ball $λ_{n} (B (z, ρ)) = ω_{n - 1} ρ^{n} / n$ with $0 < ω_{n - 1} = σ (S^{n - 1}) < \infty$ ; hence $λ_{n} (B (z, ρ)) / λ_{n} (B (z^{'}, ρ^{'})) = (ρ / ρ^{'})^{n}$ and every ball has positive finite measure (Sphere and ball measures scale in Rn, Euclidean balls have positive finite Lebesgue measure).

[F3]

A measure is countably additive on pairwise disjoint measurable sets and monotone under inclusion (Measures on sigma-algebras, Measures are monotone).

[F4]

The curve $γ$ is uniformly continuous on $[0, 1]$ . Indeed, for each $t$ continuity gives a radius $a_{t} > 0$ such that $∣ s - t ∣ < 2 a_{t}$ implies $∣ γ (s) - γ (t) ∣ < ε / 2$ . Compactness gives a finite subcover of the intervals $∣ s - t ∣ < a_{t}$ ; the minimum of its $a_{t}$ is positive. If two parameters are closer than this minimum, place the first in a covering interval and apply the two continuity bounds to obtain image distance $< ε$ .

Proof

technique · direct

1.1F1givenalgebra

Setup. By [F1] fix a John curve $γ$ from $x$ to $x_{0}$ ; the John inequality at $t = 1$ gives $∣ x - x_{0} ∣ \leq 4 c_{J} r_{0}$ . If $x \in B (x_{0}, 2 r_{0})$ we use the explicit geometric chain below; if $x \notin B (x_{0}, 2 r_{0})$ we use the recursive construction below along the John curve. In both cases all constants below depend only on $n$ and $c_{J}$ , and we collect them at the end into a single $M$ . Also $x \notin B (x_{0}, 2 r_{0})$ implies $∣ x - x_{0} ∣ \geq 2 r_{0} > r_{0}$ , so $x \notin B_{0}$ in that case.

1.2F2F3givenalgebra

The near case $x \in B (x_{0}, 2 r_{0})$ . Put $d : = ∣ x - x_{0} ∣ \leq 2 r_{0}$ . If $d > 0$ put $x_{i} : = x + 2^{- i} (x_{0} - x)$ and $r_{i} : = 2^{- i - 1} d$ for $i \geq 0$ ; if $d = 0$ fix the first standard basis vector $e_{1}$ and put $x_{i} : = x_{0} + 2^{- i} (r_{0} / 2) e_{1}$ , $r_{i} : = 2^{- i - 2} r_{0}$ . In both cases $r_{i + 1} = r_{i} / 2$ , $∣ x_{i} - x ∣ = 2 r_{i}$ , $∣ x_{i} - x_{i + 1} ∣ = r_{i}$ , $r_{i} \leq r_{0}$ and $∣ x_{i} - x_{0} ∣ \leq 2 r_{0}$ , so $dist (x_{i}, \partial Ω) \geq 4 r_{0} - ∣ x_{i} - x_{0} ∣ \geq 2 r_{0} > r_{i}$ and $B_{i} = B (x_{i}, r_{i}) \subseteq Ω$ . Consecutive balls: $B_{i} \cup B_{i + 1} \subseteq B (x_{i}, \frac{3}{2} r_{i})$ , because a point of $B_{i + 1}$ is within $∣ x_{i} - x_{i + 1} ∣ + r_{i + 1} = r_{i} + r_{i} / 2$ of $x_{i}$ , while the ball of radius $r_{i} / 4$ centred at the point of the segment from $x_{i}$ to $x_{i + 1}$ at distance $3 r_{i} / 4$ from $x_{i}$ lies in $B_{i} \cap B_{i + 1}$ (its centre is at distance $3 r_{i} / 4 < r_{i}$ from $x_{i}$ and at distance $r_{i} / 4 < r_{i + 1} = r_{i} / 2$ from $x_{i + 1}$ ); hence $∣ B_{i} \cup B_{i + 1} ∣ \leq 6^{n} ∣ B_{i} \cap B_{i + 1} ∣$ by [F2] and [F3]. Also $dist (x, B_{i}) \leq ∣ x - x_{i} ∣ = 2 r_{i}$ , and $r_{i} \to 0$ , $x_{i} \to x$ . Finally, if $y \in B_{i}$ then $r_{i} \leq ∣ x - y ∣ \leq 3 r_{i}$ ; the radii halve at each step, so the interval $[∣ x - y ∣ / 3, ∣ x - y ∣]$ of ratio $3$ contains at most two of the numbers $r_{i}$ , and $y$ therefore belongs to at most two of the balls $B_{i}$ . Hence (1), (2) and (3) hold in the near case, with ratio $6^{n}$ and multiplicity $2$ .

1.3F1givenalgebra

The far case: the recursive construction and comparability. Assume now $x \notin B (x_{0}, 2 r_{0})$ , so $x \notin B_{0}$ and $∣ x - x_{0} ∣ \geq 2 r_{0}$ . We construct $B_{i} = B (x_{i}, r_{i})$ recursively, starting with $x_{0}$ , $B_{0} = B (x_{0}, r_{0})$ . Suppose $B_{i}$ with centre $x_{i} = γ (t_{i})$ has been constructed, $B_{i} \subseteq Ω$ , and $x \notin B_{i}$ . Put $T_{i} : = {t \in [0, t_{i}] : γ (t) \in B_{i}}$ , a nonempty set containing a relative neighbourhood of $t_{i}$ in $[0, t_{i}]$ , and define $t_{i + 1} : = \inf T_{i}$ , $x_{i + 1} : = γ (t_{i + 1})$ , $r_{i + 1} : = \frac{1}{4 c_{J}} ∣ x - x_{i + 1} ∣$ and $B_{i + 1} : = B (x_{i + 1}, r_{i + 1})$ . Then $t_{i + 1} < t_{i}$ . For every $t < t_{i + 1}$ one has $γ (t) \notin B_{i}$ , while points of $T_{i}$ arbitrarily close to $t_{i + 1}$ lie in $B_{i}$ ; by continuity $x_{i + 1} \in \partial B_{i}$ , so $∣ x_{i} - x_{i + 1} ∣ = r_{i}$ , and $x_{i + 1} \neq x$ . The John inequality [F1] at $x_{i + 1}$ gives $dist (x_{i + 1}, \partial Ω) \geq c_{J}^{- 1} ∣ x - x_{i + 1} ∣ = 4 r_{i + 1} > r_{i + 1}$ , so $B_{i + 1} \subseteq Ω$ ; and $x \notin B_{i + 1}$ because $∣ x - x_{i + 1} ∣ = 4 c_{J} r_{i + 1} > r_{i + 1}$ as $c_{J} \geq 1$ . For the first transition, $∣ x_{0} - x_{1} ∣ = r_{0}$ and $∣ x - x_{0} ∣ \geq 2 r_{0}$ , so $∣ x - x_{1} ∣ \geq ∣ x - x_{0} ∣ - r_{0} \geq r_{0}$ ; also [F1] gives $∣ x - x_{0} ∣ \leq 4 c_{J} r_{0}$ , hence $∣ x - x_{1} ∣ \leq (4 c_{J} + 1) r_{0}$ . Since $r_{1} = ∣ x - x_{1} ∣ / (4 c_{J})$ , this yields $r_{0} / (4 c_{J}) \leq r_{1} \leq (1 + 1 / (4 c_{J})) r_{0}$ . For every $i \geq 1$ , the defining identity $∣ x - x_{i} ∣ = 4 c_{J} r_{i}$ and $∣ x_{i} - x_{i + 1} ∣ = r_{i}$ give $(1 - 1 / (4 c_{J})) r_{i} \leq r_{i + 1} \leq (1 + 1 / (4 c_{J})) r_{i}$ . Thus consecutive radii are comparable with ratio at most $5 / 4$ from the second transition onward, and $∣ x_{i} - x_{i + 1} ∣ = r_{i}$ is comparable to both radii for $i \geq 1$ .

2.1F1F4step 1.3givenalgebra

The far case: limit properties. The times $t_{i}$ are strictly decreasing in $[0, 1]$ , so the intervals $[t_{i + 1}, t_{i}]$ are pairwise disjoint and $\sum_{i} (t_{i} - t_{i + 1}) \leq 1$ . If $r_{i} \geq ε > 0$ for an index $i$ , then $∣ x_{i} - x_{i + 1} ∣ = r_{i} \geq ε$ , and uniform continuity [F4] of $γ$ on $[0, 1]$ gives $t_{i} - t_{i + 1} \geq δ (ε) > 0$ ; since the intervals are disjoint, only finitely many indices satisfy $r_{i} \geq ε$ . Hence $r_{i} \to 0$ ; and since $∣ x - x_{i} ∣ = 4 c_{J} r_{i}$ for every $i \geq 1$ , also $x_{i} \to x$ . Consequently $dist (x, B_{i}) \leq ∣ x - x_{i} ∣ = 4 c_{J} r_{i}$ for $i \geq 1$ , while for $i = 0$ we have $dist (x, B_{0}) \leq ∣ x - x_{0} ∣ - r_{0} \leq 4 c_{J} r_{0}$ . This gives (2) in the far case.

3.1F2F3step 1.3step 2.1algebra

The far case: multiplicity. Suppose $y$ belongs to $B_{i_{1}} \cap \dots \cap B_{i_{k}}$ with $i_{1} < \dots < i_{k}$ . Since $y \in B_{i_{j}}$ and $∣ x - x_{i_{j}} ∣ = 4 c_{J} r_{i_{j}}$ for $i_{j} \geq 1$ (while for $i_{j} = 0$ one has $∣ x - x_{0} ∣ \leq 4 c_{J} r_{0}$ and $∣ x - y ∣ \geq ∣ x - x_{0} ∣ - r_{0} \geq r_{0}$ ), the triangle inequality gives $c_{1} r_{i_{j}} \leq ∣ x - y ∣ \leq c_{2} r_{i_{j}}$ for constants $c_{1}, c_{2}$ depending only on $c_{J}$ : the upper bound is $∣ x - y ∣ \leq (4 c_{J} + 1) r_{i_{j}}$ , and the lower bound is $∣ x - y ∣ \geq (4 c_{J} - 1) r_{i_{j}}$ for $i_{j} \geq 1$ , while for $i_{j} = 0$ one has $r_{0} \leq ∣ x - y ∣ \leq (4 c_{J} + 1) r_{0}$ , so $r_{0} \geq ∣ x - y ∣ / (4 c_{J} + 1)$ and $∣ x - y ∣ \geq r_{0} / (4 c_{J} + 1) \cdot 1$ , which is the same shape with adjusted constants. Hence all the radii $r_{i_{j}}$ are comparable to $∣ x - y ∣$ . For $j < m$ one has $x_{i_{m}} \notin B_{i_{j}}$ , because $t_{i_{m}} \leq t_{i_{j} + 1}$ and $γ (t) \notin B_{i_{j}}$ for every $t < t_{i_{j} + 1}$ by step 1.3; hence $∣ x_{i_{j}} - x_{i_{m}} ∣ \geq r_{i_{j}}$ , while $∣ x_{i_{j}} - x_{i_{m}} ∣ \leq ∣ x_{i_{j}} - x ∣ + ∣ x - x_{i_{m}} ∣ \leq 4 c_{J} (r_{i_{j}} + r_{i_{m}})$ . So the $k$ centres have pairwise distances between $c_{1}^{'} ∣ x - y ∣$ and $c_{2}^{'} ∣ x - y ∣$ with constants depending only on $c_{J}$ . If $y = x$ this is impossible, because $∣ x - x_{i} ∣ = 4 c_{J} r_{i} > r_{i}$ for $i \geq 1$ and $∣ x - x_{0} ∣ \geq 2 r_{0} > r_{0}$ for $i = 0$ , so $x$ lies in no $B_{i}$ . The balls $B (x_{i_{j}}, c_{1}^{'} ∣ x - y ∣ / 3)$ are pairwise disjoint and all lie in $B (x_{i_{1}}, (c_{2}^{'} + c_{1}^{'} / 3) ∣ x - y ∣)$ , so by [F2] and [F3], $k (c_{1}^{'} / 3)^{n} \leq (c_{2}^{'} + c_{1}^{'} / 3)^{n}$ , an explicit bound $N (n, c_{J})$ .

4.1step 1.2step 1.3step 2.1step 3.1givenalgebra∎

Assembly. In the near case step 1.2 gives (1), (2) and (3) with constants depending only on $n$ : ratio at most $6^{n}$ , $dist (x, B_{i}) \leq 2 r_{i}$ , $r_{i} \to 0$ , $x_{i} \to x$ , and multiplicity $2$ . In the far case, the first pair has a separate overlap bound: the ball from step 1.3 of radius $r_{1} / 2 \geq r_{0} / (8 c_{J})$ , centred halfway from $x_{1}$ toward $x_{0}$ by $r_{1} / 2$ , lies in $B_{0} \cap B_{1}$ , while $B_{0} \cup B_{1} \subseteq B (x_{0}, r_{0} + r_{1})$ and $r_{1} \leq (1 + 1 / (4 c_{J})) r_{0}$ , so $∣ B_{0} \cup B_{1} ∣ \leq (16 c_{J} + 2)^{n} ∣ B_{0} \cap B_{1} ∣$ . For $i \geq 1$ , step 1.3 gives $r_{i + 1} \geq (3 / 4) r_{i}$ and centre separation $r_{i}$ ; the midpoint ball of radius $r_{i} / 4$ lies in $B_{i} \cap B_{i + 1}$ , while the union is contained in $B (x_{i}, 3 r_{i})$ , giving ratio at most $12^{n}$ . Step 2.1 gives $dist (x, B_{i}) \leq 4 c_{J} r_{i}$ , $r_{i} \to 0$ and $x_{i} \to x$ ; and step 3.1 gives multiplicity at most $N (n, c_{J})$ . Taking $M \geq \max (6^{n}, (16 c_{J} + 2)^{n}, 12^{n}, 4 c_{J}, N (n, c_{J}), 2, 1)$ completes the proof. A John curve was selected once, for the given $x$ , in step 1.1.

Source notes

The construction and properties are Kinnunen's, printed pp. 141-142: the radius $r_{i + 1} = ∣ x - x_{i + 1} ∣ / (4 c_{J})$ at the last exit point of the ball, the comparability of consecutive radii and centre distances, the packing bound on centres with pairwise comparable distances, and the terminal convergence $r_{i} \to 0$ , $x_{i} \to x$ . Kinnunen leaves the case $x \in B (x_{0}, 2 r_{0})$ as an exercise; the explicit overlapping geometric chain of step 1.2 supplies it. The roles of the constants are kept separate: the John inequality is used only to put every ball inside $Ω$ and to bound $∣ x - x_{0} ∣$ , the packing bound uses only the comparability of the radii to $∣ x - y ∣$ , and no monotonicity of the radii is claimed.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

The truncated Riesz kernel is bounded on $L^{p}$ of a bounded set

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 2$ , let $z_{0} \in R^{n}$ , $ρ > 0$ , and let $Ω \subseteq B (z_{0}, ρ)$ be measurable, $1 \leq p < \infty$ and $f \in L^{p} (Ω; K)$ . Then $∥ \int_{Ω} ∣ x - y ∣^{1 - n} ∣ f (y) ∣ d y ∥_{L^{p} (Ω)} \leq C (n) ρ ∥ f ∥_{L^{p} (Ω)}$ . In particular, for a bounded John domain the John condition gives $diam Ω \leq c ∣ Ω ∣^{1 / n}$ , so the bound holds with coefficient $C (n, c_{J}) ∣ Ω ∣^{1 / n}$ .

Facts & Assumptions

Given: Countable Choice; $n \geq 2$ ; $z_{0} \in R^{n}$ , $ρ > 0$ ; a measurable $Ω \subseteq B (z_{0}, ρ)$ ; $1 \leq p < \infty$ ; $f \in L^{p} (Ω; K)$ ; and, for the final claim, a bounded John domain $Ω$ with distinguished point $x_{0}$ and admissible constant $c_{J} \geq 1$ .

[F1]

The polar surface measure is $σ (E) = n λ_{n} ({r ω : ω \in E, 0 < r \leq 1})$ on Borel $E \subseteq S^{n - 1}$ (The polar surface set function on the unit sphere).

[F2]

Polar coordinates: $\int_{R^{n}} h d λ_{n} = \int_{0}^{\infty} \int_{S^{n - 1}} h (r ω) r^{n - 1} d σ (ω) d r$ for nonnegative Borel $h$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

[F3]

Every Euclidean ball has positive finite Lebesgue measure (Euclidean balls have positive finite Lebesgue measure).

[F4]

Minkowski's integral inequality: $∥ \int_{Y} ∣ F (\cdot, y) ∣ d ν (y) ∥_{L^{p} (X)} \leq \int_{Y} ∥ F (\cdot, y) ∥_{L^{p} (X)} d ν (y)$ for measurable $F$ with the right side finite (Minkowski's integral inequality).

[F5]

Tonelli-Fubini on completed sigma-finite products gives measurability and equality of the nonnegative iterated integrals (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability). Lebesgue translation invariance gives $∥ g (\cdot - z) ∥_{p} = ∥ g ∥_{p}$ (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).

[F6]

A measure is countably additive on pairwise disjoint measurable sets, and monotone under inclusion (Measures on sigma-algebras, Measures are monotone).

[F7]

A John domain with admissible constant $c_{J}$ and point $x_{0}$ admits, for every $x \in Ω$ , a curve from $x$ to $x_{0}$ with $dist (γ (t), \partial Ω) \geq c_{J}^{- 1} ∣ x - γ (t) ∣$ (John domains and the John constant).

[F9]

Measurability is preimage measurability, and $L^{p}$ consists of almost-everywhere classes of measurable functions with finite norm (A measurable function between measurable spaces, The space $L^{p} (μ)$ as the quotient by null functions).

[F10]

Countable Choice, used by the cited measure-theoretic interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F11]

Under Countable Choice, every completion-measurable real function has a base-measurable representative equal to it almost everywhere (A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra); Lebesgue measure is the completion of Borel Lebesgue measure ( $L (R^{n})$ is exactly the completion of the restriction of $λ_{n}$ to the Borel sets). Applying this componentwise gives a finite Borel representative of every $L^{p} (R^{n}; K)$ class.

[F12]

Under Countable Choice, reflection in the origin preserves Lebesgue measurability and measure (For a nonzero real $c$ , dilation by $c$ multiplies Lebesgue outer measure by $∣ c ∣^{n}$ , and reflection in the origin preserves it).

Proof

technique · direct

1.1F1F2F3F10algebra

The truncated kernel has finite mass. Put $k (z) : = ∣ z ∣^{1 - n} 1_{B (0, 2 ρ)} (z)$ for $z \neq 0$ and $k (0) : = 0$ . By [F2] applied to the nonnegative Borel function $k$ and by [F1], [F3], $∥ k ∥_{L^{1} (R^{n})} = \int_{S^{n - 1}} \int_{0}^{2 ρ} t^{1 - n} t^{n - 1} d t d σ (ω) = 2 ρ σ (S^{n - 1}) = 2 ρ n λ_{n} (B (0, 1)) = : C_{0} (n) ρ$ , so $∥ k ∥_{1}$ is finite and equals a dimension-only multiple of $ρ$ .

1.2F2F6F7algebra

John-domain volume control. Put $a : = dist (x_{0}, \partial Ω) > 0$ . The ball $B (x_{0}, a)$ lies in $Ω$ : a segment from $x_{0}$ to any point outside $Ω$ first meets the boundary, so an outside point cannot be closer than $a$ . Polar coordinates [F2] therefore give $∣ Ω ∣ \geq σ (S^{n - 1}) a^{n} / n$ . Evaluating [F7] at the endpoint of the curve gives $∣ x - x_{0} ∣ \leq c_{J} a$ for every $x \in Ω$ , hence $diam Ω \leq 2 c_{J} a \leq 2 c_{J} (n / σ (S^{n - 1}))^{1 / n} ∣ Ω ∣^{1 / n}$ .

2.1F2F3F4F5F11F12step 1.1algebra

The convolution bound. For $g \in L^{p} (R^{n}; K)$ , choose a finite Borel representative $g_{0}$ by [F11] (replace any infinite values on a Borel null set by zero). The function $F_{0} (x, z) = k (z) g_{0} (x - z)$ is Borel, hence measurable for the product of the Lebesgue sigma-algebras. The two Lebesgue spaces are sigma-finite, being exhausted by bounded balls of finite measure [F2, F3]. Translation invariance [F5] and step 1.1 give $\int ∥ F_{0} (\cdot, z) ∥_{p} d z = \int k (z) ∥ g_{0} (\cdot - z) ∥_{p} d z = ∥ k ∥_{1} ∥ g ∥_{p} < \infty$ . Minkowski [F4] therefore shows that the absolute integral is finite almost everywhere and $∥ T g_{0} ∥_{p} \leq ∥ k ∥_{1} ∥ g ∥_{p}$ , where $T g_{0} (x) = \int k (z) g_{0} (x - z) d z$ . This also defines $T g$ for any Lebesgue representative $g$ : for each fixed $x$ , the exceptional set of $z$ is the translate and reflection $x - N$ of the null set $N$ where $g \neq g_{0}$ , and has measure zero by reflection invariance [F12] and translation invariance [F5]. Thus the section integrals agree wherever finite, and the measurable almost-everywhere representative $T g_{0}$ supplies $∥ T g ∥_{p} \leq ∥ k ∥_{1} ∥ g ∥_{p}$ .

2.2step 1.2algebra

In particular step 1.2 gives $diam Ω \leq 2 c (n, c_{J}) ∣ Ω ∣^{1 / n}$ with $c (n, c_{J}) : = c_{J} (n / σ (S^{n - 1}))^{1 / n}$ ; taking a supremum does not require that a farthest point exist.

3.1F5F9F12step 2.1givenalgebra

The bound on $Ω$ for a set inside a ball. Extend $f$ by zero to $R^{n}$ and put $g : = ∣ f ∣ 1_{Ω}$ , a measurable function with $∥ g ∥_{L^{p} (R^{n})} = ∥ f ∥_{L^{p} (Ω)}$ by [F9]. By translation and reflection invariance [F5, F12], the substitution $y = x - z$ gives $T g (x) = \int k (x - y) g (y) d y$ wherever finite. Since $x, y \in Ω \subseteq B (z_{0}, ρ)$ implies $∣ x - y ∣ \leq 2 ρ$ , for almost every $x \in Ω$ one has $T g (x) = \int_{Ω} ∣ x - y ∣^{1 - n} ∣ f (y) ∣ d y$ : the kernel truncation in the convolution is inactive exactly on the pairs with $∣ x - y ∣ < 2 ρ$ . Hence, by step 2.1, $∥ \int_{Ω} ∣ x - y ∣^{1 - n} ∣ f (y) ∣ d y ∥_{L^{p} (Ω)} \leq ∥ T g ∥_{L^{p} (R^{n})} \leq ∥ k ∥_{L^{1}} ∥ f ∥_{L^{p} (Ω)} \leq C (n) ρ ∥ f ∥_{L^{p} (Ω)}$ .

4.1step 3.1step 2.2algebra∎

The John-domain form of the bound. Apply step 3.1 with $z_{0} : = x_{0}$ and $ρ : = 2 c (n, c_{J}) ∣ Ω ∣^{1 / n}$ , which is admissible by step 2.2 because then $Ω \subseteq B (x_{0}, ρ)$ . The resulting coefficient is $C (n) ρ = C (n, c_{J}) ∣ Ω ∣^{1 / n}$ .

Source notes

Kinnunen proves Lemma 5.15 by Holder and Fubini, using the kernel integral estimate of Lemma 5.14; the proof above instead uses Minkowski's integral inequality for the truncated radial kernel, which gives the bound on every $L^{p}$ with the single constant $∥ k ∥_{1} = C (n) ρ$ and avoids interpolation. The John-domain volume estimate follows from the interior ball at the distinguished point and the endpoint John inequality.

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The mean-zero Poincare inequality on bounded John domains

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be a bounded John domain with admissible constant $c_{J}$ , and let $1 \leq p < \infty$ . There is a constant $C (n, p, c_{J})$ with $∥ u - u_{Ω} ∥_{L^{p} (Ω)} \leq C (n, p, c_{J}) diam (Ω) ∥ D u ∥_{L^{p} (Ω)}$ for every $u \in W^{1, p} (Ω; K)$ , where $u_{Ω} = ∣ Ω ∣^{- 1} \int_{Ω} u$ . The factor $diam (Ω)$ is necessary: for the function $u (x) = x_{1}$ on a ball of radius $R$ with the centre as distinguished point, the ratio $∥ u - u_{B} ∥_{L^{p} (B)} / ∥ D u ∥_{L^{p} (B)}$ grows linearly in $R$ , while the John constant of that pair is $1$ for every $R$ .

Facts & Assumptions

Given: The Axiom of Choice, whose Countable-Choice consequence is used for the measure-theoretic interfaces; a bounded John domain $Ω$ with distinguished point $x_{0}$ and admissible constant $c_{J}$ ; $1 \leq p < \infty$ ; a field $K$ ; and a class $u \in W^{1, p} (Ω; K)$ .

[F1]

The John chain lemma: with $r_{0} = \frac{1}{4} dist (x_{0}, \partial Ω)$ and $B_{0} = B (x_{0}, r_{0})$ there is $M = M (n, c_{J})$ such that for every $x \in Ω$ there are balls $B_{i} = B (x_{i}, r_{i}) \subseteq Ω$ with $∣ B_{i} \cup B_{i + 1} ∣ \leq M ∣ B_{i} \cap B_{i + 1} ∣$ , $dist (x, B_{i}) \leq M r_{i}$ , $r_{i} \to 0$ , $x_{i} \to x$ , and multiplicity at most $M$ (Bounded-overlap ball chains in a bounded John domain; $x_{0}, c_{J}$ as in John domains and the John constant).

[F2]

Ball oscillation and ball Poincare: for $v \in W^{1, p} (B)$ , $∥ v - v_{B} ∥_{L^{p} (B)} \leq C_{1} (n, p) r ∥ D v ∥_{L^{p} (B)}$ and $\int_{B} ∣ v - v_{B} ∣ \leq C_{2} (n) r \int_{B} ∣ D v ∣$ for a ball $B = B (x, r)$ (Poincare inequality on a ball, Ball-mean oscillation bound by the Riesz potential of the gradient).

[F3]

At almost every Lebesgue point $x$ of $u$ , the centered averages $A_{r} u (x)$ converge to $u (x)$ as $r ↓ 0$ (Lebesgue differentiation theorem on $R^{n}$ , The average of a locally integrable function over a Euclidean ball).

[F4]

The truncated Riesz kernel bound: for measurable $Ω^{'} \subseteq B (z_{0}, ρ)$ and $f \in L^{p} (Ω^{'})$ , $∥ \int_{Ω^{'}} ∣ x - y ∣^{1 - n} ∣ f (y) ∣ d y ∥_{L^{p} (Ω^{'})} \leq C (n) ρ ∥ f ∥_{L^{p} (Ω^{'})}$ (The truncated Riesz kernel is bounded on $L^{p}$ of a bounded set).

[F5]

Holder's inequality and Tonelli's theorem for nonnegative functions on sigma-finite products (Holder's inequality for integrals, including the endpoint cases, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product); $W^{1, p}$ consists of $L^{p}$ classes with gradient in $L^{p}$ (Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions).

[F6]

Linear substitution scales Lebesgue measure by the absolute determinant (A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not); smooth classical derivatives are weak derivatives (Classical derivatives agree with weak derivatives), and balls have positive finite measure scaling with the nth power of the radius (Sphere and ball measures scale in Rn).

Proof

technique · direct

1.1F1F2F3algebra

Lebesgue points and the near case. Extend $u$ by zero outside $Ω$ ; it is locally integrable. The cited differentiation theorem implies $A_{s} (∣ u - u (x) ∣) (x) \to 0$ at almost every $x$ : apply it simultaneously to $∣ u - a ∣$ for the countable dense set $a \in Q$ (or $Q + i Q$ ), and bound ${lim sup}_{s ↓ 0} A_{s} (∣ u - u (x) ∣) (x) \leq 2 ∣ a - u (x) ∣$ ; let $a \to u (x)$ . Write $B_{0} = B (x_{0}, r_{0})$ for the fixed central ball and $B_{*} : = B (x_{0}, 2 r_{0}) \subset Ω$ . For almost every $x \in B_{*}$ , [F2] gives $∣ u (x) - u_{B_{*}} ∣ \leq C (n) \int_{B_{*}} ∣ D u (y) ∣ ∣ x - y ∣^{1 - n} d y$ . Also $∣ u_{B_{*}} - u_{B_{0}} ∣ \leq ∣ B_{0} ∣^{- 1} \int_{B_{*}} ∣ u - u_{B_{*}} ∣ \leq C (n) r_{0}^{1 - n} \int_{B_{*}} ∣ D u ∣$ by the $L^{1}$ ball Poincare estimate. Since $∣ x - y ∣ \leq 4 r_{0}$ for $x, y \in B_{*}$ , this last bound is at most $C (n) \int_{B_{*}} ∣ D u (y) ∣ ∣ x - y ∣^{1 - n} d y$ . Thus $∣ u (x) - u_{B_{0}} ∣ \leq C (n) \int_{Ω} ∣ D u (y) ∣ ∣ x - y ∣^{1 - n} d y$ in the near case, with the same fixed $B_{0}$ for all $x$ .

1.2F1F2F3givenalgebra

Telescoping along the chain. Fix a Lebesgue point $x \notin B (x_{0}, 2 r_{0})$ of $u$ and a chain ${B_{i} = B (x_{i}, r_{i})}$ from [F1], with overlap and distance constant $M$ . Since $dist (x, B_{i}) \leq M r_{i}$ , we have $∣ x_{i} - x ∣ \leq (M + 1) r_{i}$ and hence $B_{i} \subseteq B (x, (M + 2) r_{i})$ . The volume ratio is $∣ B (x, (M + 2) r_{i}) ∣ / ∣ B_{i} ∣ = (M + 2)^{n}$ , so [F3] gives $∣ u_{B_{i}} - u (x) ∣ \leq \frac{1}{∣ B_{i} ∣} \int_{B_{i}} ∣ u (y) - u (x) ∣ d y \leq (M + 2)^{n} A_{(M + 2) r_{i}} (∣ u - u (x) ∣) (x) ⟶ 0.$ Thus $u_{B_{i}} \to u (x)$ , and $∣ u (x) - u_{B_{0}} ∣ \leq \sum_{i \geq 0} ∣ u_{B_{i}} - u_{B_{i + 1}} ∣$ . Writing each difference of means as an average over the intersection and using $∣ B_{i} ∣, ∣ B_{i + 1} ∣ \leq M ∣ B_{i} \cap B_{i + 1} ∣$ from [F1], $∣ u_{B_{i}} - u_{B_{i + 1}} ∣ \leq M (\frac{\int_{B_{i}} ∣ u - u_{B_{i}} ∣}{∣ B_{i} ∣} + \frac{\int_{B_{i + 1}} ∣ u - u_{B_{i + 1}} ∣}{∣ B_{i + 1} ∣}) .$ Applying the $L^{1}$ ball Poincare inequality [F2] on each ball and using the radius comparability supplied by [F1] gives $∣ u_{B_{i}} - u_{B_{i + 1}} ∣ \leq C_{3} (n, c_{J}) \sum_{j \in {i, i + 1}} r_{j} \frac{\int_{B_{j}} ∣ D u ∣}{∣ B_{j} ∣} .$

2.1F1F5step 1.1step 1.2givenalgebra

The potential bound. From step 1.2, $∣ u (x) - u_{B_{0}} ∣ \leq C_{3} \sum_{i} r_{i} \int_{B_{i}} ∣ D u ∣ / ∣ B_{i} ∣$ (each ball counted with a bounded number of neighbours with comparable radii, the constant absorbed into $C_{3}$ ). For $y \in B_{i}$ the chain property $dist (x, B_{i}) \leq M r_{i}$ gives $∣ x - y ∣ \leq (M + 2) r_{i}$ , hence $∣ B_{i} ∣ = ω_{n - 1} r_{i}^{n} / n$ and $r_{i} / ∣ B_{i} ∣ = c (n) r_{i}^{1 - n} \leq c (n, M) ∣ x - y ∣^{1 - n}$ ; summing over $i$ and using the multiplicity bound of [F1], $\sum_{i} r_{i} \int_{B_{i}} ∣ D u ∣ / ∣ B_{i} ∣ \leq c (n, M) \int_{Ω} ∣ D u (y) ∣ ∣ x - y ∣^{1 - n} N (x, y) d y \leq c (n, M) \int_{Ω} ∣ D u (y) ∣ ∣ x - y ∣^{1 - n} d y$ with $N (x, y) \leq M$ the number of balls containing $y$ ; the exchange of sum and integral is Tonelli [F5]. The index $i = 0$ needs the separate bound $r_{0} \int_{B_{0}} ∣ D u ∣ / ∣ B_{0} ∣ \leq c (n, c_{J}) \int_{B_{0}} ∣ D u (y) ∣ ∣ x - y ∣^{1 - n} d y$ : the John condition at $x$ gives $diam Ω \leq 8 c_{J} r_{0}$ , so $∣ x - y ∣ \leq 8 c_{J} r_{0}$ for $y \in B_{0}$ and $r_{0}^{1 - n} \leq (8 c_{J})^{n - 1} ∣ x - y ∣^{1 - n}$ . Together with step 1.1, this gives $∣ u (x) - u_{B_{0}} ∣ \leq C_{4} (n, c_{J}) \int_{Ω} ∣ D u (y) ∣ ∣ x - y ∣^{1 - n} d y$ for almost every $x \in Ω$ .

3.1F1F4F5step 2.1givenalgebra

$L^{p}$ norms and the mean. Take $L^{p} (Ω)$ norms in step 2.1 and apply the truncated kernel bound [F4] with $Ω^{'} = Ω \subseteq B (x_{0}, diam Ω)$ and $f = ∣ D u ∣$ : $∥ u - u_{B_{0}} ∥_{L^{p} (Ω)} \leq C_{5} (n, c_{J}) diam (Ω) ∥ D u ∥_{L^{p} (Ω)}$ . Since $u_{B_{0}}$ is a constant, $u_{Ω} - u_{B_{0}} = ∣ Ω ∣^{- 1} \int_{Ω} (u - u_{B_{0}})$ , so by Holder [F5] $∣ u_{B_{0}} - u_{Ω} ∣ \leq ∣ Ω ∣^{- 1} \int_{Ω} ∣ u - u_{B_{0}} ∣ \leq ∣ Ω ∣^{- 1 / p} ∥ u - u_{B_{0}} ∥_{L^{p} (Ω)}$ . Hence $∥ u - u_{Ω} ∥_{L^{p} (Ω)} \leq ∥ u - u_{B_{0}} ∥_{L^{p} (Ω)} + ∣ Ω ∣^{1 / p} ∣ u_{B_{0}} - u_{Ω} ∣ \leq 2 ∥ u - u_{B_{0}} ∥_{L^{p} (Ω)} \leq C (n, p, c_{J}) diam (Ω) ∥ D u ∥_{L^{p} (Ω)}$ with $C (n, p, c_{J}) : = 2 C_{5} (n, c_{J})$ .

4.1F1F6algebra∎

Necessity of length scaling. On $B = B (0, R)$ take $u (x) = x_{1}$ . By [F6] its weak gradient is $e_{1}$ , and reflection in the first coordinate gives $u_{B} = 0$ . Substituting $x = R y$ yields $∥ u ∥_{L^{p} (B)} = R^{1 + n / p} (\int_{B (0, 1)} ∣ y_{1} ∣^{p} d y)^{1 / p}$ and $∥ D u ∥_{L^{p} (B)} = R^{n / p} ∣ B (0, 1) ∣^{1 / p}$ . The first integral is finite and positive, since the unit ball contains a ball on which $∣ y_{1} ∣$ is bounded below by a positive number. Their ratio is therefore $c (n, p) R$ with $c (n, p) > 0$ . The radial segment from any $x$ to $0$ satisfies $R - ∣ γ (t) ∣ \geq ∣ x ∣ - ∣ γ (t) ∣ = ∣ x - γ (t) ∣$ , so this distinguished pair admits John constant $1$ for every $R$ . Thus no dimension-and-John-constant bound can omit the length factor.

Source notes

Kinnunen proves the Sobolev-Poincare inequality on John domains (Theorem 5.33, printed pp. 141-143) by exactly this chaining: the telescoping over $∣ u_{B_{i}} - u_{B_{i + 1}} ∣$ , the ball Poincare inequality, the comparison $r_{i} \approx ∣ x - y ∣$ on $B_{i}$ , the multiplicity bound, and the truncated-kernel estimate. The present item states the $L^{p}$ (rather than $L^{p^{*}}$ ) mean-zero form, which is what the surrounding page promises; the chaining argument is the same, and no Sobolev exponent is used. The endpoint index $i = 0$ is absorbed with the John bound $diam Ω \leq 8 c_{J} r_{0}$ .

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The Poincare inequality with a positive-measure zero set

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $n \geq 2$ , let $Ω$ be a bounded John domain with admissible constant $c_{J}$ , let $1 \leq p < \infty$ , and let $A \subseteq Ω$ be measurable with $∣ A ∣ \geq γ ∣ Ω ∣$ for some $γ \in (0, 1]$ . If $u \in W^{1, p} (Ω; K)$ vanishes almost everywhere on $A$ , then $∥ u ∥_{L^{p} (Ω)} \leq C (n, p, c_{J}, γ) diam (Ω) ∥ D u ∥_{L^{p} (Ω)} .$

Facts & Assumptions

Given: The Axiom of Choice; a bounded John domain $Ω$ with constant $c_{J}$ (John domains and the John constant); $1 \leq p < \infty$ ; a measurable $A \subseteq Ω$ with $∣ A ∣ \geq γ ∣ Ω ∣$ , $γ \in (0, 1]$ ; and a class $u \in W^{1, p} (Ω; K)$ vanishing almost everywhere on $A$ .

[F1]

The mean-zero Poincare inequality on the John domain: $∥ w - w_{Ω} ∥_{L^{p} (Ω)} \leq C_{1} (n, p, c_{J}) diam (Ω) ∥ D w ∥_{L^{p} (Ω)}$ for every $w \in W^{1, p} (Ω; K)$ , where $w_{Ω} = ∣ Ω ∣^{- 1} \int_{Ω} w$ (The mean-zero Poincare inequality on bounded John domains, The average of a locally integrable function over a Euclidean ball).

[F2]

Weak derivatives are linear: $D (w - c) = D w$ for every constant $c$ (Linearity, locality, and commutation of weak derivatives); $W^{1, p}$ consists of $L^{p}$ classes with weak gradient in $L^{p}$ , and $L^{p}$ is a space of almost-everywhere classes (Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions).

[F3]

Holder's inequality: $\int_{A} ∣ v ∣ \leq ∣ A ∣^{1 - 1 / p} ∥ v ∥_{L^{p} (Ω)}$ for the finite measure set $A$ (Holder's inequality for integrals, including the endpoint cases).

Proof

technique · direct

1.1F1F2givenalgebra

The mean-zero part. Since constants have zero weak derivative, $v : = u - u_{Ω}$ satisfies $v \in W^{1, p} (Ω; K)$ and $D v = D u$ by [F2]; by [F1], $∥ v ∥_{L^{p} (Ω)} \leq C_{1} (n, p, c_{J}) diam (Ω) ∥ D u ∥_{L^{p} (Ω)}$ . Also $v = u - u_{Ω}$ almost everywhere on $A$ , so $u$ vanishes there if and only if $v = - u_{Ω}$ there.

2.1F3step 1.1givenalgebra

Recovering the mean from the zero set. Because $u = 0$ almost everywhere on $A$ and $∣ A ∣ \leq ∣ Ω ∣ < \infty$ , $\int_{A} v = \int_{A} (u - u_{Ω}) = - ∣ A ∣ u_{Ω}$ ; hence $∣ A ∣ ∣ u_{Ω} ∣ \leq \int_{A} ∣ v ∣ \leq ∣ A ∣^{1 - 1 / p} ∥ v ∥_{L^{p} (Ω)} \leq ∣ A ∣^{1 - 1 / p} C_{1} diam (Ω) ∥ D u ∥_{L^{p} (Ω)}$ by [F3] and step 1.1, and therefore $∣ u_{Ω} ∣ \leq ∣ A ∣^{- 1 / p} C_{1} diam (Ω) ∥ D u ∥_{L^{p} (Ω)} \leq (γ ∣ Ω ∣)^{- 1 / p} C_{1} diam (Ω) ∥ D u ∥_{L^{p} (Ω)}$ .

3.1step 1.1step 2.1givenalgebra∎

Conclusion. By the triangle inequality and steps 1.1 and 2.1, $∥ u ∥_{L^{p} (Ω)} \leq ∥ u - u_{Ω} ∥_{L^{p} (Ω)} + ∣ Ω ∣^{1 / p} ∣ u_{Ω} ∣ \leq C_{1} diam (Ω) ∥ D u ∥_{L^{p} (Ω)} + C_{1} γ^{- 1 / p} diam (Ω) ∥ D u ∥_{L^{p} (Ω)} = C (n, p, c_{J}, γ) diam (Ω) ∥ D u ∥_{L^{p} (Ω)}$ with $C (n, p, c_{J}, γ) : = C_{1} (1 + γ^{- 1 / p})$ , which is the asserted inequality.

Source notes

Kinnunen's Remark 3.20 records the zero-set variant: a function vanishing on a set of positive measure can be normalised without the mean, and the mean itself is controlled by the amount of mass on the complement. The proof above implements that normalisation: the mean-zero inequality controls $u - u_{Ω}$ , and the value $u_{Ω}$ is recovered from the zero set by integrating $u - u_{Ω}$ over $A$ . The exponent $- 1 / p$ in the mean bound is what produces the constant $γ^{- 1 / p}$ .

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Poincare-Wirtinger on bounded convex domains by the direct pairwise argument

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $Ω \subseteq R^{n}$ be open, bounded, convex and nonempty, and let $1 \leq p < \infty$ . Then $∥ u - u_{Ω} ∥_{L^{p} (Ω)} \leq C (n) diam (Ω) ∥ D u ∥_{L^{p} (Ω)}$ for every $u \in W^{1, p} (Ω; K)$ .

Here $u_{Ω} = ∣ Ω ∣^{- 1} \int_{Ω} u d x$ is the mean of $u$ over $Ω$ , which is well defined because $Ω$ is nonempty open and bounded. The proof below gives the explicit choice $C (n) = \frac{2 (2^{n} - 1)}{n},$ which depends only on the dimension; no dependence on $p$ , on the shape of $Ω$ , on $∣ Ω ∣$ or on the regularity of $\partial Ω$ is used.

Facts & Assumptions

Given: The Axiom of Choice; an integer $n \geq 1$ ; an open, bounded, convex, nonempty set $Ω \subseteq R^{n}$ with $d : = diam (Ω)$ ; an exponent $1 \leq p < \infty$ ; a field $K \in {R, C}$ ; and a class $u \in W^{1, p} (Ω; K)$ .

[F1]

$W^{1, p} (Ω; K)$ consists of the $L^{p} (Ω; K)$ classes whose weak first derivatives exist as $L^{p}$ classes, and $∥ D u ∥_{L^{p} (Ω)}$ is the $L^{p}$ norm of the weak gradient (Integer-order Sobolev spaces and their norms); an element of $L^{p}$ is an almost-everywhere equivalence class of measurable representatives (The space $L^{p} (μ)$ as the quotient by null functions).

[F2]

The Axiom of Choice is the statement that every family of nonempty sets has a choice function (The Axiom of Choice); it implies the Axiom of Countable Choice, the statement that every at most countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F3]

Under Countable Choice, for every open $U \subset \subset Ω$ the interior mollifications $u_{ε}$ of $u$ satisfy $u_{ε} \to u$ in $W^{1, p} (U; K)$ as $ε \to 0^{+}$ (Local smooth approximation in integer-order Sobolev spaces).

[F4]

On a completed sigma-finite product, nonnegative measurable functions may be integrated in either order and the iterated integrals agree, and integrable functions obey the same identity; this is Tonelli-Fubini (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability).

[F5]

Jensen's inequality: for a probability space, an integrable real function $f$ with values in an interval $I$ and a convex $φ$ on $I$ such that $φ \circ f \in L^{1} (P)$ , one has $φ (\int f d P) \leq \int φ (f) d P$ (Jensen's integral inequality for a probability measure).

[F6]

Holder's inequality: for conjugate exponents $p, q$ and measurable real $f, g$ in the corresponding $L$ -spaces, $\int ∣ f g ∣ d μ \leq ∥ f ∥_{p} ∥ g ∥_{q}$ (Holder's inequality for integrals, including the endpoint cases).

[F7]

For a $C^{1}$ diffeomorphism $T : U \to V$ between open subsets of $R^{m}$ and every nonnegative Borel $h : V \to [0, \infty]$ , $\int_{V} h (y) d y = \int_{U} h (T (x)) ∣ \det D T (x) ∣ d x$ , with equality in $[0, \infty]$ and the convention $0 \cdot \infty = 0$ (Borel change of variables from the compact-support formula and Radon uniqueness).

[F8]

If $f : [a, b] \to R^{m}$ is differentiable with integrable derivative then $\int_{a}^{b} f^{'} = f (b) - f (a)$ (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz); if $g \circ f$ is formed from totally differentiable maps then $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F9]

Every Euclidean ball has positive finite Lebesgue measure (Euclidean balls have positive finite Lebesgue measure), a measure is monotone under inclusion (Measures are monotone), and every bounded subset of $R^{n}$ has finite outer measure (Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure).

[F10]

Dominated convergence applies to integrable majorants (Dominated convergence).

Proof

technique · direct

1.1F1F6F9givenalgebra

Means and integrability. Since $Ω$ is nonempty and open it contains a Euclidean ball $B \subseteq Ω$ , and [F9] gives $0 < λ (B) \leq λ (Ω)$ by monotonicity; since $Ω$ is bounded, [F9] gives $∣ Ω ∣ < \infty$ . Thus $0 < ∣ Ω ∣ < \infty$ and $u_{Ω}$ is defined once $u$ is integrable. For $1 < p < \infty$ Holder's inequality [F6] with $g = 1_{Ω}$ gives $∥ u ∥_{1} \leq ∣ Ω ∣^{1 - 1 / p} ∥ u ∥_{p} < \infty$ , and for $p = 1$ integrability is immediate; in both cases $u \in L^{1} (Ω; K)$ and $u_{Ω}$ is an element of $K$ , understood componentwise when $K = C ≅ R^{2}$ .

1.2F5F8givenalgebra

The smooth segment inequality. Let $V \subseteq Ω$ be open and convex and let $v \in C^{\infty} (V; K) \cap W^{1, p} (V; K)$ . Fix $x, y \in V$ and put $γ (t) = x + t (y - x)$ for $t \in [0, 1]$ ; convexity gives $γ ([0, 1]) \subseteq V$ . By the chain rule [F8], $v \circ γ$ is differentiable with $(v \circ γ)^{'} (t) = D v (γ (t)) (y - x)$ , and the vector-valued fundamental theorem [F8] gives $v (x) - v (y) = \int_{0}^{1} D v (γ (t)) (x - y) d t$ . Hence $∣ v (x) - v (y) ∣ \leq ∣ x - y ∣ \int_{0}^{1} ∣ D v (γ (t)) ∣ d t$ , and, since $D v$ is continuous on the compact segment, $∣ D v \circ γ ∣^{p}$ is integrable. Jensen's inequality [F5] applied to the probability measure $d t$ on $[0, 1]$ and the convex function $φ (s) = s^{p}$ on $[0, \infty)$ yields $∣ v (x) - v (y) ∣^{p} \leq ∣ x - y ∣^{p} \int_{0}^{1} ∣ D v (γ (t)) ∣^{p} d t$ .

2.1F4F7step 1.2algebra

The first half of the substitution. For $t \in [1 / 2, 1]$ and fixed $x \in V$ , the affine diffeomorphism $y \mapsto z = (1 - t) x + t y$ has image $(1 - t) x + t V \subseteq V$ and inverse Jacobian $t^{- n}$ . Bound $∣ x - y ∣ \leq d$ before substituting in [F7]; this gives $\int_{V} ∣ x - y ∣^{p} ∣ D v ((1 - t) x + t y) ∣^{p} d y \leq d^{p} t^{- n} \int_{V} ∣ D v ∣^{p}$ . Integrating over $x \in V$ and $t \in [1 / 2, 1]$ yields $c_{n} d^{p} ∣ V ∣ \int_{V} ∣ D v ∣^{p}$ , where $c_{n} : = \int_{1 / 2}^{1} t^{- n} d t$ .

2.2F4F7step 1.2algebra

The second half. For $t \in [0, 1 / 2]$ and fixed $y \in V$ , substitute $z = (1 - t) x + t y \in V$ in the $x$ integral. The bound $∣ x - y ∣ \leq d$ and [F7] give $\int_{V} ∣ x - y ∣^{p} ∣ D v ((1 - t) x + t y) ∣^{p} d x \leq d^{p} (1 - t)^{- n} \int_{V} ∣ D v ∣^{p}$ . Integration over $y$ and $t$ , with $s = 1 - t$ , gives the same $c_{n} d^{p} ∣ V ∣ \int_{V} ∣ D v ∣^{p}$ .

3.1F4F5step 2.1step 2.2algebra

The mean-zero bound for smooth functions. Adding steps 2.1 and 2.2 and using [F4] to identify the iterated integral over $V \times V \times [0, 1]$ of the nonnegative integrand with the sum of its two halves, $\int_{V} \int_{V} ∣ v (x) - v (y) ∣^{p} d x d y \leq 2 c_{n} d^{p} ∣ V ∣ \int_{V} ∣ D v ∣^{p} .$ The normalized Lebesgue measure $d y / ∣ V ∣$ is a probability measure, so using $∣ \int h ∣ \leq \int ∣ h ∣$ and scalar Jensen [F5] for $s \mapsto s^{p}$ , with $v_{V} = ∣ V ∣^{- 1} \int_{V} v$ , the required integrability holds because $v \in L^{p} (V)$ implies $∣ v (x) - v (\cdot) ∣^{p} \in L^{1} (V)$ for every fixed $x$ ; hence one has $∣ v (x) - v_{V} ∣^{p} \leq ∣ V ∣^{- 1} \int_{V} ∣ v (x) - v (y) ∣^{p} d y$ for every $x \in V$ , and integrating in $x$ yields $\int_{V} ∣ v - v_{V} ∣^{p} \leq 2 c_{n} d^{p} \int_{V} ∣ D v ∣^{p} .$

4.1F1F2F3step 3.1givenalgebra

Passage to $W^{1, p}$ and to the whole of $Ω$ . Choose open convex sets $V_{1} \subseteq V_{2} \subseteq \dots \subset \subset Ω$ with $⋃_{j} V_{j} = Ω$ , for instance $V_{j} = {x \in Ω : dist (x, R^{n} ∖ Ω) > 1 / j} \cap B (0, j)$ . Fix $j$ ; by [F3] the mollifications $u_{ε}$ of $u$ converge to $u$ in $W^{1, p} (V_{j}; K)$ as $ε \to 0^{+}$ , and each $u_{ε}$ is smooth on a neighbourhood of $V_{j}$ . Step 3.1 applied to $v = u_{ε}$ on the convex set $V_{j}$ , followed by the limits $∥ u_{ε} - u ∥_{L^{p} (V_{j})} \to 0$ , $∥ D u_{ε} - D u ∥_{L^{p} (V_{j})} \to 0$ and $(u_{ε})_{V_{j}} \to u_{V_{j}}$ as $ε \to 0^{+}$ , gives $\int_{V_{j}} ∣ u - u_{V_{j}} ∣^{p} \leq 2 c_{n} d^{p} \int_{V_{j}} ∣ D u ∣^{p} \leq 2 c_{n} d^{p} \int_{Ω} ∣ D u ∣^{p} .$

5.1F10step 4.1algebra∎

Exhaustion and the constant. Discard the finitely many empty $V_{j}$ . Since $V_{j} ↑ Ω$ and $u \in L^{1}$ , dominated convergence [F10] gives $∣ V_{j} ∣ \to ∣ Ω ∣$ and $u_{V_{j}} \to u_{Ω}$ . The means are bounded; hence $1_{V_{j}} ∣ u - u_{V_{j}} ∣^{p}$ is dominated by $2^{p - 1} (∣ u ∣^{p} + \sup_{j} ∣ u_{V_{j}} ∣^{p})$ on the finite-measure set $Ω$ . By [F10] it converges in integral to $∣ u - u_{Ω} ∣^{p}$ , and similarly $\int_{V_{j}} ∣ D u ∣^{p} \to \int_{Ω} ∣ D u ∣^{p}$ . Step 4.1 yields $∥ u - u_{Ω} ∥_{p} \leq (2 c_{n})^{1 / p} d ∥ D u ∥_{p}$ . For $1 / 2 \leq t \leq 1$ , $t^{- n} \geq 1$ and $t^{- n} \leq t^{- n - 1}$ , so $1 \leq 2 c_{n} \leq 2 \int_{1 / 2}^{1} t^{- n - 1} d t = 2 (2^{n} - 1) / n$ . Thus $(2 c_{n})^{1 / p} \leq 2 c_{n} \leq 2 (2^{n} - 1) / n$ for every $p \geq 1$ , proving the stated dimension-only constant.

Source notes

The computation follows Kinnunen's ball proof of the pointwise oscillation estimate and the Poincare inequality, printed pp. 133-136, with the segment argument of Lemma 5.22: the ball is replaced by the convex set, polar coordinates and the maximal function are not needed, and the two halves of the parameter interval carry the substitution from the moving interior point to a fixed one. The constant is not claimed to be sharp; the dimension-only bound is the conclusion used here.

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Sobolev embedding on bounded extension domains for $p < n$

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be a bounded $W^{1, p}$ -extension domain with a bounded extension operator $E$ , and let $1 \leq p < n$ , $p \leq q \leq p^{*} = \frac{n p}{n - p}$ . Then $W^{1, p} (Ω) ↪ L^{q} (Ω)$ continuously: $∥ u ∥_{L^{q} (Ω)} \leq C (n, p, q, Ω) ∥ u ∥_{W^{1, p} (Ω)} (u \in W^{1, p} (Ω; K)) .$

Facts & Assumptions

Given: The Axiom of Choice; $n \geq 2$ ; a bounded extension domain $Ω$ with a bounded extension operator $E$ and operator norm $∥ E ∥$ (Sobolev extension domains and extension operators); $1 \leq p < n$ ; $p \leq q \leq p^{*}$ ; a field $K$ .

[F1]

For $1 < p < n$ , the whole-space Gagliardo-Nirenberg-Sobolev inequality holds (The Gagliardo-Nirenberg-Sobolev inequality for $1 < p < n$ ). For $p = 1$ , compactly supported smooth density (Compactly supported smooth functions are dense in W^{k,p}(R^n)) extends The p=1 Gagliardo-Nirenberg-Sobolev inequality to $W^{1, 1} (R^{n})$ : the smooth estimate on differences gives an $L^{n / (n - 1)}$ Cauchy sequence, completeness gives its limit, and successive almost-everywhere subsequences in that space and in $L^{1}$ identify the limit with the Sobolev class (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , Complex Lp completeness and almost-everywhere subsequences). Thus $∥ F ∥_{p^{*}} \leq C (n, p) ∥ D F ∥_{p}$ for every $1 \leq p < n$ (The Sobolev conjugate exponent and the scaling identity).

[F2]

The transfer corollary: if $N$ is a whole-space functional with $N_{Ω} (F ∣_{Ω}) \leq N (F)$ and $N (F) \leq C ∥ F ∥_{W^{1, p} (R^{n})}$ , then $N_{Ω} (u) \leq C ∥ E ∥ ∥ u ∥_{W^{1, p} (Ω)}$ for all $u$ in $W^{1, p} (Ω)$ (Whole-space inequalities transfer through a Sobolev extension).

[F3]

Holder's inequality gives the $L^{q}$ interpolation bound $∥ v ∥_{L^{q} (Ω)} \leq ∥ v ∥_{L^{p} (Ω)}^{θ} ∥ v ∥_{L^{p^{*}} (Ω)}^{1 - θ}$ whenever $1 \leq p \leq q \leq p^{*} < \infty$ and $\frac{1}{q} = \frac{θ}{p} + \frac{1 - θ}{p^{*}}$ , on a finite measure set (Holder's inequality for integrals, including the endpoint cases).

[F4]

$W^{1, p}$ consists of $L^{p}$ classes with weak gradient in $L^{p}$ , and $∥ v ∥_{L^{p} (Ω)} \leq ∥ v ∥_{W^{1, p} (Ω)}$ (Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions); every bounded $C^{k}$ domain supplies an admissible extension operator through the extension theorem (Bounded C^k domains admit integer-order Sobolev extension).

Proof

technique · direct

1.1F1F2givenalgebra

The endpoint $q = p^{*}$ . Apply the transfer corollary [F2] with $N (F) = ∥ F ∥_{L^{p^{*}} (R^{n})}$ , which satisfies the two hypotheses by [F1], enlarging $C (n, p)$ by the finite-dimensional comparison between the Euclidean gradient and the coordinate Sobolev norm: $∥ F ∥_{L^{p^{*}} (R^{n})} \leq C (n, p) ∥ F ∥_{W^{1, p} (R^{n})}$ and, since $E u$ restricts to $u$ almost everywhere, $∥ u ∥_{L^{p^{*}} (Ω)} = ∥ (E u) ∣_{Ω} ∥_{L^{p^{*}} (Ω)} \leq ∥ E u ∥_{L^{p^{*}} (R^{n})}$ . Hence $∥ u ∥_{L^{p^{*}} (Ω)} \leq C (n, p) ∥ E ∥ ∥ u ∥_{W^{1, p} (Ω)}$ .

2.1F3F4step 1.1givenalgebra

Intermediate exponents. For $p \leq q \leq p^{*}$ write $\frac{1}{q} = \frac{θ}{p} + \frac{1 - θ}{p^{*}}$ with $θ = \frac{1 / q - 1 / p^{*}}{1 / p - 1 / p^{*}} \in [0, 1]$ (at $q = p$ take $θ = 1$ , at $q = p^{*}$ take $θ = 0$ ). By [F3], $∥ u ∥_{L^{q} (Ω)} \leq ∥ u ∥_{L^{p} (Ω)}^{θ} ∥ u ∥_{L^{p^{*}} (Ω)}^{1 - θ} \leq \max (1, C (n, p) ∥ E ∥) ∥ u ∥_{W^{1, p} (Ω)}$ using $∥ u ∥_{L^{p}} \leq ∥ u ∥_{W^{1, p}}$ from [F4] and step 1.1; the case $q = p$ is the same inequality with $θ = 1$ .

3.1F4step 1.1step 2.1givenalgebra∎

The constant and the domain class. The constant obtained depends only on $n, p, q$ and $∥ E ∥$ , hence only on $n, p, q, Ω$ for a fixed extension domain; every bounded $C^{k}$ domain, $k \geq 1$ , supplies an admissible $E$ through [F4], so the embedding applies in particular to that class.

Source notes

The endpoint case is the whole-space Sobolev inequality transferred through a bounded extension operator, Kinnunen's Theorem 3.43 (printed pp. 84–85) and Definition 3.42; the intermediate exponents are the standard Holder interpolation between $L^{p}$ and $L^{p^{*}}$ on the finite measure set $Ω$ . The constant is not asserted to be uniform over all extension domains, matching the transfer corollary's dependence on $∥ E ∥$ .

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Sobolev-Poincare on bounded connected extension domains

Statement

Assume the Axiom of Choice (and hence Countable Choice and Dependent Choice). Let $n \geq 2$ , $1 < p < n$ , $p^{*} = \frac{n p}{n - p}$ , let $K \in {R, C}$ , and let $Ω \subset R^{n}$ be a nonempty bounded connected $W^{1, p}$ -extension domain. There is $C = C (n, p, Ω, K)$ such that every $u \in W^{1, p} (Ω; K)$ satisfies $∥ u - u_{Ω} ∥_{L^{p^{*}} (Ω)} \leq C ∥ D u ∥_{L^{p} (Ω)}, u_{Ω} = ∣ Ω ∣^{- 1} \int_{Ω} u .$ The domain constant is not uniform over arbitrary extension domains.

Facts & Assumptions

Given: The Axiom of Choice, hence Countable Choice and Dependent Choice; $n \geq 2$ ; $1 < p < n$ ; a nonempty bounded connected $W^{1, p}$ -extension domain $Ω$ (Sobolev extension domains and extension operators); a field $K$ ; and a class $u \in W^{1, p} (Ω; K)$ .

[F1]

The local mean-zero estimate on bounded connected extension domains: there is $C_{P} = C_{P} (n, p, Ω, K)$ with $∥ w - w_{Ω} ∥_{L^{p} (Ω)} \leq C_{P} ∥ D w ∥_{L^{p} (Ω)}$ for every $w \in W^{1, p} (Ω; K)$ (Mean-zero Poincare estimate on bounded connected extension domains below the dimension).

[F2]

Constants have zero weak derivative and weak derivatives are linear, so $D (w - c) = D w$ (Linearity, locality, and commutation of weak derivatives); $W^{1, p}$ consists of $L^{p}$ classes with weak gradient in $L^{p}$ (Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions).

[F3]

The extension-domain embedding: for $1 \leq p < n$ and $p \leq q \leq p^{*}$ , $∥ w ∥_{L^{q} (Ω)} \leq C_{E} (n, p, q, Ω) ∥ w ∥_{W^{1, p} (Ω)}$ (Sobolev embedding on bounded extension domains for $p < n$ , The Sobolev conjugate exponent and the scaling identity).

[F4]

On the finite measure set $Ω$ , Holder's inequality gives $w \in L^{1} (Ω)$ for every $w \in L^{p} (Ω)$ with $∥ w ∥_{1} \leq ∣ Ω ∣^{1 - 1 / p} ∥ w ∥_{p}$ , so the mean $u_{Ω}$ is defined (Holder's inequality for integrals, including the endpoint cases).

Proof

technique · direct

1.1F1F2F4givenalgebra

Centering and the local estimate. By [F4] the mean $u_{Ω}$ is a well-defined scalar; put $v : = u - u_{Ω}$ . By [F2], $v \in W^{1, p} (Ω; K)$ with $D v = D u$ and $v_{Ω} = 0$ , and by [F1] applied to $v$ , $∥ v ∥_{L^{p} (Ω)} \leq C_{P} ∥ D u ∥_{L^{p} (Ω)}$ .

2.1F3step 1.1givenalgebra∎

The critical exponent. Apply the extension-domain embedding [F3] to $v$ with $q = p^{*}$ : $∥ v ∥_{L^{p^{*}} (Ω)} \leq C_{E} ∥ v ∥_{W^{1, p} (Ω)}$ ; since $∥ v ∥_{W^{1, p} (Ω)}$ is, up to a dimension-only factor, $∥ v ∥_{L^{p} (Ω)} + ∥ D v ∥_{L^{p} (Ω)} \leq (1 + C_{P}) ∥ D u ∥_{L^{p} (Ω)}$ by step 1.1, renaming the product constant gives $∥ u - u_{Ω} ∥_{L^{p^{*}} (Ω)} \leq C (n, p, Ω, K) ∥ D u ∥_{L^{p} (Ω)}$ , which is the asserted inequality.

Source notes

Kinnunen's Theorem 3.47 is the mean-zero $L^{p^{*}}$ estimate, whose proof first establishes the mean-zero $L^{p}$ estimate on bounded connected extension domains, proved there by Rellich compactness; the local item cited as [F1] supplies it directly with the extension-cutoff-mollification and Arzela-Ascoli argument on this page. The step from the $L^{p}$ mean-zero estimate to the critical exponent is the extension-domain embedding, exactly as Kinnunen combines Theorem 3.47 with the Sobolev embedding. The constant depends on the extension operator through $C_{E}$ ; no uniformity over all extension domains is claimed, matching the statement.

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The critical Sobolev embedding into every finite $L^{q}$

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $n \geq 2$ and let $Ω \subseteq R^{n}$ be nonempty, open, bounded and a $W^{1, p}$ -extension domain for every $p \in (n / 2, n)$ (the extension operator may depend on $p$ ). For every $1 \leq q < \infty$ there is $C (n, q, Ω)$ with $∥ u ∥_{L^{q} (Ω)} \leq C (n, q, Ω) ∥ u ∥_{W^{1, n} (Ω)} (u \in W^{1, n} (Ω; K));$ that is, $W^{1, n} (Ω) ↪ L^{q} (Ω)$ continuously for every finite $q$ . The constant necessarily blows up as $q \to \infty$ , so no $L^{\infty}$ bound is asserted.

Facts & Assumptions

Given: The Axiom of Choice; $n \geq 2$ ; a bounded nonempty open set $Ω$ that is a $W^{1, p}$ -extension domain for every $p \in (n / 2, n)$ , with the operator allowed to depend on $p$ (Sobolev extension domains and extension operators); $1 \leq q < \infty$ ; and a class $u \in W^{1, n} (Ω; K)$ .

[F1]

Holder's inequality on the finite measure set $Ω$ : for $1 \leq a \leq b < \infty$ , $∥ w ∥_{L^{a} (Ω)} \leq ∣ Ω ∣^{1 / a - 1 / b} ∥ w ∥_{L^{b} (Ω)}$ (Holder's inequality for integrals, including the endpoint cases).

[F2]

Subcritical Sobolev embedding: for $1 \leq p < n$ and $p \leq r \leq p^{*} = \frac{n p}{n - p}$ , $∥ w ∥_{L^{r} (Ω)} \leq C (n, p, r, Ω) ∥ w ∥_{W^{1, p} (Ω)}$ on a bounded extension domain (Sobolev embedding on bounded extension domains for $p < n$ ).

[F3]

$W^{1, n}$ consists of the $L^{n}$ classes with all first weak derivatives in $L^{n}$ , and the Sobolev norm is the $ℓ^{n}$ norm of the component $L^{n}$ norms (Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions, The Sobolev conjugate exponent and the scaling identity).

[F4]

If $\overline{B (a, 2 R)} \subset Ω$ , there is $η \in C_{c}^{\infty} (B (a, 2 R))$ equal to $1$ on $B (a, R)$ (A Euclidean bump for a compact set inside an open set). Polar coordinates give the radial integral formula (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, The polar surface set function on the unit sphere), and weak derivatives are defined by integration by parts against compactly supported smooth tests (Weak derivative of a locally integrable function).

Proof

technique · direct

1.1F1F3givenalgebra

The case $q \leq n$ . By [F1] with $a = q$ and $b = n$ , $∥ u ∥_{L^{q} (Ω)} \leq ∣ Ω ∣^{1 / q - 1 / n} ∥ u ∥_{L^{n} (Ω)} \leq ∣ Ω ∣^{1 / q - 1 / n} ∥ u ∥_{W^{1, n} (Ω)}$ .

1.2F1F2F3givenalgebra

The case $q > n$ . Put $s : = \frac{n q}{n + q}$ . Then $n / 2 < s < n$ and $s^{*} = \frac{n s}{n - s} = q$ by [F3]. By the domain hypothesis, $Ω$ is a $W^{1, s}$ -extension domain, so the subcritical embedding [F2] with $p = s$ and $r = q = s^{*}$ gives $∥ u ∥_{L^{q} (Ω)} \leq C (n, s, q, Ω) ∥ u ∥_{W^{1, s} (Ω)}$ . Holder [F1] applied to each component with $a = s < b = n$ gives $∥ D^{α} u ∥_{L^{s}} \leq ∣ Ω ∣^{1 / s - 1 / n} ∥ D^{α} u ∥_{L^{n}}$ for $∣ α ∣ \leq 1$ ; summing over the finitely many multi-indices yields $∥ u ∥_{W^{1, s} (Ω)} \leq C (∣ Ω ∣, n, s) ∥ u ∥_{W^{1, n} (Ω)}$ . Combining the two estimates proves the case $q > n$ ; since $s = n q / (n + q)$ , the resulting constant depends only on $n, q, Ω$ .

2.1F3F4step 1.1step 1.2givenalgebra∎

Conclusion. Steps 1.1 and 1.2 cover $q \leq n$ and $q > n$ , so every finite $q$ is covered with a constant depending only on $n, q, Ω$ . To see that these constants cannot remain bounded as $q \to \infty$ , choose $a \in Ω$ and $R > 0$ with $\overline{B (a, 2 R)} \subset Ω$ , and choose $η$ as in [F4]. Define $w (x) = η (x) \log \log (1 + R / ∣ x - a ∣)$ for $x \neq a$ , assigning any finite value at $a$ . If $r = ∣ x - a ∣ \leq R / 2$ , then $∣ D \log \log (1 + R / r) ∣ = \frac{R}{r (r + R) \log (1 + R / r)} \leq \frac{1}{r \log (R / r)} .$ Thus polar coordinates [F4] give $\int_{B (a, R / 2)} ∣ D w ∣^{n} d x \leq C \int_{0}^{R / 2} \frac{d r}{r (\log (R / r))^{n}} < \infty$ for $n \geq 2$ ; also $w \in L^{n}$ near $a$ because $\log \log (1 + R / r) = O (r^{- 1 / 2})$ and the polar $L^{n}$ integral of $r^{- 1 / 2}$ converges for $n \geq 2$ , and on the rest of its compact support $w$ is smooth with bounded derivatives. For every coordinate line with nonzero transverse displacement from $a$ , $w$ is smooth and compactly supported on that line. Apply the fundamental theorem (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz) to $w$ times a test function. The transverse singleton of excluded lines is null since $n \geq 2$ , and $w, D w \in L^{n} \subseteq L^{1}$ on their bounded support. Fubini (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability) therefore integrates the section identities to the weak derivative identity, proving $w \in W^{1, n} (Ω)$ . It is essentially unbounded on every neighbourhood of $a$ : for every $M > 0$ it exceeds $M$ on a punctured ball about $a$ , so $E_{M}$ has positive measure. If $q_{j} \to \infty$ and admissible constants satisfied $C (q_{j}) \leq C_{*}$ , then for each $M > 0$ , the set $E_{M} = {x : ∣ w (x) ∣ > M}$ has positive measure and $C_{*} ∥ w ∥_{W^{1, n} (Ω)} \geq ∥ w ∥_{L^{q_{j}} (Ω)} \geq M ∣ E_{M} ∣^{1 / q_{j}} .$ Letting $j \to \infty$ gives $C_{*} ∥ w ∥_{W^{1, n}} \geq M$ for every $M$ , a contradiction. Hence the best constants necessarily diverge as $q \to \infty$ , and no $L^{\infty}$ endpoint is asserted.

Source notes

Kinnunen's Remark 3.16 and Hunter's discussion at $p = n$ record the critical embedding $W^{1, n} ↪ L^{q}$ for finite $q$ and the failure of the $L^{\infty}$ endpoint. The proof above reduces $q > n$ to the subcritical embedding with source exponent $s = n q / (n + q) \in (n / 2, n)$ and uses Holder on the finite measure domain to compare $W^{1, s}$ with $W^{1, n}$ . The all-exponents hypothesis supplies exactly this $W^{1, s}$ extension operator for each finite $q > n$ ; the case $q \leq n$ is direct Holder.

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Ball-mean oscillation bound by the Riesz potential of the gradient

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 2$ , let $1 \leq p < \infty$ , let $B (x, r) \subseteq R^{n}$ be a ball, and let $u \in W^{1, p} (B (x, r); K)$ with ball average $u_{B (x, r)}$ . Then $∣ u (z) - u_{B (x, r)} ∣ \leq C (n) \int_{B (x, r)} ∣ D u (y) ∣ ∣ z - y ∣^{1 - n} d y$ for almost every $z \in B (x, r)$ ; here $∣ D u ∣$ is the Euclidean norm of the weak gradient and $C (n)$ depends only on $n$ .

Facts & Assumptions

Given: Countable Choice; $n \geq 2$ ; $x \in R^{n}$ and $r > 0$ ; $1 \leq p < \infty$ ; a field $K \in {R, C}$ ; and a class $u \in W^{1, p} (B (x, r); K)$ .

[F1]

The polar surface measure is normalized by $σ (E) = n λ_{n} ({t ω : ω \in E, 0 < t \leq 1})$ on Borel $E \subseteq S^{n - 1}$ , and for nonnegative Borel $h$ one has $\int_{R^{n}} h d λ_{n} = \int_{0}^{\infty} \int_{S^{n - 1}} h (t ω) t^{n - 1} d σ (ω) d t$ (The polar surface set function on the unit sphere, Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

[F2]

For every ball $λ_{n} (B (z, ρ)) = σ (S^{n - 1}) ρ^{n} / n$ , and this is positive and finite (Sphere and ball measures scale in Rn).

[F3]

If a curve is composed from differentiable maps then the chain rule computes its derivative, and a differentiable curve with integrable derivative satisfies the fundamental theorem of calculus (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz).

[F4]

On completed sigma-finite products nonnegative measurable functions may be integrated in either order (Tonelli-Fubini), and an invertible linear map scales Lebesgue measure by $∣ \det ∣$ (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability, A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not).

[F5]

Truncated Riesz kernel bound: for measurable $Ω \subseteq B (z_{0}, ρ)$ and $f \in L^{p} (Ω; K)$ , $∥ \int_{Ω} ∣ x - y ∣^{1 - n} ∣ f (y) ∣ d y ∥_{L^{p} (Ω)} \leq C (n) ρ ∥ f ∥_{L^{p} (Ω)}$ (The truncated Riesz kernel is bounded on $L^{p}$ of a bounded set).

[F6]

Interior mollifications converge to $u$ in $W^{1, p} (U)$ on every $U ⋐ B (x, r)$ (Local smooth approximation in integer-order Sobolev spaces). Norm convergence has an almost-everywhere convergent subsequence (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , Complex Lp completeness and almost-everywhere subsequences). Dominated convergence applies to integrable majorants (Dominated convergence).

[F7]

The ball average $u_{B} = ∣ B ∣^{- 1} \int_{B} u$ is the normalized integral of the class, and $W^{1, p}$ consists of the $L^{p}$ classes with weak gradient in $L^{p}$ (The average of a locally integrable function over a Euclidean ball, Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions).

[F8]

On a finite measure space $L^{p}$ includes into $L^{1}$ , so $u_{i} \to u$ in $L^{p}$ implies $(u_{i})_{B} \to u_{B}$ (Finite-measure $L^{r}$ includes into $L^{p}$ for $p < r$ ).

[F9]

The Axiom of Countable Choice is available and is used through the cited measure-theoretic and approximation interfaces (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1F1F3F4givenalgebra

Spherical oscillation bound for a smooth function. Let $v \in C^{\infty} (R^{n}; K)$ , $B : = B (x, r)$ , and fix $z \in B$ and $ρ > 0$ . For $y \in B$ the chain rule and the fundamental theorem [F3] applied to $t \mapsto v (t y + (1 - t) z)$ give $v (y) - v (z) = \int_{0}^{1} D v (t y + (1 - t) z) \cdot (y - z) d t$ , hence $∣ v (y) - v (z) ∣ \leq ∣ y - z ∣ \int_{0}^{1} ∣ D v (t y + (1 - t) z) ∣ d t$ . Writing $S_{ρ} : = \partial B (z, ρ)$ for the sphere equipped with its surface measure, and using that the homothety $y \mapsto t y + (1 - t) z$ maps $S_{ρ}$ onto $S_{t ρ}$ with surface element scaled by $t^{n - 1}$ ; its image of $B \cap S_{ρ}$ is contained in $B \cap S_{t ρ}$ by convexity, so enlargement gives the inequality below (the surface measures are the polar measures of cones, so this is the linear change-of-variables property of [F1] and [F4]), $\int_{B \cap S_{ρ}} ∣ v (y) - v (z) ∣ d S (y) \leq ρ \int_{0}^{1} \int_{B \cap S_{ρ}} ∣ D v (t y + (1 - t) z) ∣ d S (y) d t \leq ρ \int_{0}^{1} t^{1 - n} \int_{B \cap S_{t ρ}} ∣ D v (w) ∣ d S (w) d t$ . Since $∣ w - z ∣ = t ρ$ on $S_{t ρ}$ , the inner integral equals $ρ^{n - 1} t^{n - 1} \int_{B \cap S_{t ρ}} ∣ D v (w) ∣ ∣ z - w ∣^{1 - n} d S (w)$ ; substituting $s = t ρ$ , so that $d t = d s / ρ$ , the last display becomes $ρ^{n - 1} \int_{0}^{ρ} \int_{B \cap S_{s}} ∣ D v (w) ∣ ∣ z - w ∣^{1 - n} d S (w) d s = ρ^{n - 1} \int_{B \cap B (z, ρ)} ∣ D v (w) ∣ ∣ z - w ∣^{1 - n} d w$ , the final equality being the polar-coordinate formula [F1] for the nonnegative function $w \mapsto ∣ D v (w) ∣ ∣ z - w ∣^{1 - n}$ on $B \cap B (z, ρ)$ (whose singularity at $w = z$ is integrable; a single point is null).

2.1F1F2F7step 1.1algebra

The oscillation bound for a smooth function. Let $v \in C^{\infty} (R^{n}; K)$ and keep $B = B (x, r)$ . Since $u_{B}$ is the normalized integral, $∣ v (z) - v_{B} ∣ = ∣ B ∣^{- 1} ∣ \int_{B} (v (z) - v (y)) d y ∣$ ; writing the integral over $B$ in polar coordinates around $z$ and using $B \subseteq B (z, 2 r)$ , step 1.1 gives $∣ v (z) - v_{B} ∣ \leq ∣ B ∣^{- 1} \int_{0}^{2 r} \int_{B \cap S_{ρ}} ∣ v (y) - v (z) ∣ d S (y) d ρ \leq ∣ B ∣^{- 1} \int_{0}^{2 r} ρ^{n - 1} d ρ \int_{B} ∣ D v (y) ∣ ∣ z - y ∣^{1 - n} d y$ for every $z \in B$ . By [F2], $∣ B ∣ = σ (S^{n - 1}) r^{n} / n$ and $\int_{0}^{2 r} ρ^{n - 1} d ρ = (2 r)^{n} / n$ , so $∣ B ∣^{- 1} \int_{0}^{2 r} ρ^{n - 1} d ρ = 2^{n} / σ (S^{n - 1}) = : c (n)$ ; hence $∣ v (z) - v_{B} ∣ \leq c (n) \int_{B} ∣ D v (y) ∣ ∣ z - y ∣^{1 - n} d y$ for every $z \in B$ .

3.1F5F6F8F9step 2.1algebra

First pass to the Sobolev class on an inner ball $B_{j} : = B (x, r (1 - 1 / j))$ , $j \geq 2$ . Its closure lies in $B$ , so [F6] supplies smooth mollifications $u_{i} \to u$ in $W^{1, p} (B_{j})$ . By [F8] their $B_{j}$ means converge. Apply step 2.1 on $B_{j}$ . The potential operator on $B_{j}$ satisfies [F5], and $∣ \int_{B_{j}} ∣ D u_{i} (y) ∣ ∣ z - y ∣^{1 - n} d y - \int_{B_{j}} ∣ D u (y) ∣ ∣ z - y ∣^{1 - n} d y ∣ \leq \int_{B_{j}} ∣ D u_{i} - D u ∣ (y) ∣ z - y ∣^{1 - n} d y$ tends to zero in $L^{p} (B_{j})$ by [F5]. Successive almost-everywhere subsequences from [F6] for $u_{i}$ and these potentials therefore give $∣ u (z) - u_{B_{j}} ∣ \leq c (n) \int_{B_{j}} ∣ D u (y) ∣ ∣ z - y ∣^{1 - n} d y$ for almost every $z \in B_{j}$ .

4.1F6F8step 3.1algebra∎

Take the union of the countably many exceptional null sets from step 3.1. For $z$ outside this union, $z \in B_{j}$ for every sufficiently large $j$ , and each right side is at most $c (n) \int_{B} ∣ D u (y) ∣ ∣ z - y ∣^{1 - n} d y$ . Since $u \in L^{1} (B)$ , dominated convergence [F6] gives $u_{B_{j}} \to u_{B}$ . Letting $j \to \infty$ proves the asserted inequality on $B$ , without any approximation claim at its boundary.

Source notes

The computation is Kinnunen's Lemma 5.22, printed pp. 133-135: the spherical change of variables $w = t y + (1 - t) z$ , the radius substitution $s = t ρ$ and the final polar-coordinate identity are reproduced with their justification, and the explicit constant $2^{n} / σ (S^{n - 1})$ is recorded. Kinnunen states the lemma for $C^{1} (R^{n})$ functions and then passes to $W_{l o c}^{1, p}$ by mollification and the $L^{p}$ bound for the Riesz potential of the gradient; the passage above uses the library's interior mollification and its truncated-kernel bound, which already carries the John-domain rescaling used later on the companion page.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Morrey's inequality for $p > n$

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $n \geq 2$ , $n < p < \infty$ , let $Ω \subseteq R^{n}$ be open and let $u \in W^{1, p} (Ω; K)$ . Then $u$ has a continuous representative $u^{*}$ , and for every ball $B (x, r)$ with $B (x, 2 r) ⋐ Ω$ one has $∣ u^{*} (z) - u^{*} (y) ∣ \leq C (n, p) ∣ z - y ∣^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, 2 r))} (z, y \in B (x, r));$ equivalently $[u^{*}]_{C^{0, 1 - n / p} (B (x, r))} \leq C (n, p) ∥ D u ∥_{L^{p} (B (x, 2 r))}$ .

Facts & Assumptions

Given: The Axiom of Choice, used through the Countable-Choice interfaces of the cited measure-theoretic and approximation results; $n \geq 2$ ; $n < p < \infty$ ; an open set $Ω \subseteq R^{n}$ ; a field $K \in {R, C}$ ; and a class $u \in W^{1, p} (Ω; K)$ .

[F1]

The ball oscillation estimate: for a ball $B (x, ρ)$ and $u \in W^{1, p} (B (x, ρ))$ , $∣ u (z) - u_{B (x, ρ)} ∣ \leq C_{0} (n) \int_{B (x, ρ)} ∣ D u (y) ∣ ∣ z - y ∣^{1 - n} d y$ for almost every $z \in B (x, ρ)$ (Ball-mean oscillation bound by the Riesz potential of the gradient).

[F2]

Holder's inequality $\int ∣ f g ∣ \leq ∥ f ∥_{p} ∥ g ∥_{p^{'}}$ for conjugate exponents (Holder's inequality for integrals, including the endpoint cases), and $\int_{B (z, R)} ∣ z - y ∣^{(1 - n) p^{'}} d y = ω_{n - 1} R^{n - (n - 1) p^{'}} / (n - (n - 1) p^{'})$ with $n - (n - 1) p^{'} > 0$ because $p > n$ , while $(n - (n - 1) p^{'}) / p^{'} = 1 - n / p$ by polar coordinates (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

[F3]

The ball average is the normalized integral, and $A_{ρ} u (x) \to u (x)$ as $ρ ↓ 0$ for almost every $x$ (The average of a locally integrable function over a Euclidean ball, Lebesgue differentiation theorem on $R^{n}$ ).

[F4]

$W^{1, p}$ consists of the $L^{p}$ classes with weak gradient in $L^{p}$ ; $L^{p}$ classes are determined up to null sets; for $α = 1 - n / p \in (0, 1)$ the local Holder norm is $[v]_{C^{0, α} (B)} = \sup_{z \neq y \in B} ∣ v (z) - v (y) ∣ / ∣ z - y ∣^{α}$ (Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions, Local Hölder and scaled C-two-alpha norms on balls).

[F5]

Countable Choice is available from the Axiom of Choice and is the hypothesis of the cited differentiation interface; the cited Holder theorem requires no choice hypothesis (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1F1F2givenalgebra

Oscillation on a ball. Fix a ball $B (x, ρ) \subseteq Ω$ . By [F1], for almost every $z \in B (x, ρ)$ , $∣ u (z) - u_{B (x, ρ)} ∣ \leq C_{0} (n) \int_{B (x, ρ)} ∣ D u (y) ∣ ∣ z - y ∣^{1 - n} d y$ . Holder [F2] with exponents $p$ and $p^{'}$ bounds this by $C_{0} (n) ∥ D u ∥_{L^{p} (B (x, ρ))} (\int_{B (z, 2 ρ)} ∣ z - y ∣^{(1 - n) p^{'}} d y)^{1 / p^{'}}$ , where the integration region has been enlarged from $B (x, ρ) \subseteq B (z, 2 ρ)$ to $B (z, 2 ρ)$ . By [F2] the kernel integral equals $c (n, p) ρ^{n - (n - 1) p^{'}}$ with $n - (n - 1) p^{'} > 0$ and exponent $(n - (n - 1) p^{'}) / p^{'} = 1 - n / p$ , so $∣ u (z) - u_{B (x, ρ)} ∣ \leq C_{1} (n, p) ρ^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, ρ))}$ for almost every $z \in B (x, ρ)$ .

2.1F2F3F4F5step 1.1givenalgebra

Convergence of the ball means and the representative. Fix a ball $B (x, r)$ with $B (x, 2 r) ⋐ Ω$ and let $0 < ρ \leq σ \leq r$ . For almost every $z \in B (x, ρ)$ step 1.1 applied on $B (x, σ)$ gives $∣ u (z) - u_{B (x, σ)} ∣ \leq C_{1} σ^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, σ))} \leq C_{1} σ^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, r))}$ ; averaging in $z$ over $B (x, ρ)$ yields $∣ u_{B (x, ρ)} - u_{B (x, σ)} ∣ \leq C_{1} σ^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, r))}$ . Hence the net $(u_{B (x, ρ)})_{ρ ↓ 0}$ is Cauchy and we may define $u^{*} (x) : = \lim_{ρ ↓ 0} u_{B (x, ρ)}$ for every $x$ such that some $B (x, 2 r) ⋐ Ω$ , with $∣ u^{*} (x) - u_{B (x, ρ)} ∣ \leq C_{1} ρ^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, r))}$ for $0 < ρ \leq r$ . The zero extension of $u$ to $R^{n}$ lies in $L_{l o c}^{1} (R^{n})$ by Holder [F2] on bounded balls, since $u \in L^{p} (Ω)$ ; its sufficiently small ball means at each interior point are those of $u$ . Thus [F3] and [F4] give $u_{B (x, ρ)} \to u (x)$ for almost every $x$ , so $u^{*} = u$ almost everywhere: $u^{*}$ is a representative of the class. The Countable-Choice interface used here is supplied by [F5].

3.1F4step 1.1step 2.1givenalgebra∎

The Holder bound and continuity. Let $z, y \in B (x, r)$ and put $ℓ : = ∣ z - y ∣$ ; the case $ℓ = 0$ is trivial. If $ℓ \leq r / 2$ , apply step 1.1 on the balls $B (z, ℓ)$ and $B (y, ℓ)$ and step 2.1 on their means: $∣ u^{*} (z) - u_{B (z, ℓ)} ∣ \leq C_{1} ℓ^{1 - n / p} ∥ D u ∥_{L^{p} (B (z, ℓ))}$ and similarly at $y$ ; both balls lie in $B (x, r + ℓ) \subseteq B (x, 2 r)$ when $ℓ \leq r / 2$ , so the two norms are at most $∥ D u ∥_{L^{p} (B (x, 2 r))}$ . For the difference of the two means, both balls $B (z, ℓ)$ and $B (y, ℓ)$ lie in $B (z, 2 ℓ) \subseteq B (x, r + 2 ℓ) \subseteq B (x, 2 r)$ , and step 1.1 on $B (z, 2 ℓ)$ bounds $∣ u (a) - u (b) ∣ \leq 2 C_{1} (2 ℓ)^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, 2 r))}$ for almost every $a \in B (z, ℓ)$ , $b \in B (y, ℓ)$ ; averaging gives $∣ u_{B (z, ℓ)} - u_{B (y, ℓ)} ∣ \leq 2 C_{1} (2 ℓ)^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, 2 r))}$ . Combining the three terms, $∣ u^{*} (z) - u^{*} (y) ∣ \leq C_{2} (n, p) ℓ^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, 2 r))}$ . If $ℓ > r / 2$ , use the mean over the fixed ball $B (x, 2 r)$ : by step 1.1 on $B (x, 2 r)$ , for almost every $a \in B (x, 2 r)$ one has $∣ u (a) - u_{B (x, 2 r)} ∣ \leq C_{1} (2 r)^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, 2 r))}$ , and since $B (z, ρ) \subseteq B (x, 2 r)$ for $0 < ρ < r$ , averaging over $B (z, ρ)$ and letting $ρ ↓ 0$ gives $∣ u^{*} (z) - u_{B (x, 2 r)} ∣ \leq C_{1} (2 r)^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, 2 r))}$ , with the same bound at $y$ ; hence $∣ u^{*} (z) - u^{*} (y) ∣ \leq 2 C_{1} (2 r)^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, 2 r))} \leq C_{2}^{'} ℓ^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, 2 r))}$ because $(2 r)^{1 - n / p} \leq 4^{1 - n / p} ℓ^{1 - n / p}$ when $ℓ > r / 2$ . This proves the displayed estimate with a constant depending only on $n$ and $p$ ; since the exponent $1 - n / p$ is positive, $u^{*}$ is continuous on every ball $B (x, r)$ with $B (x, 2 r) ⋐ Ω$ , hence on all of $Ω$ , and the equivalent Holder-norm statement follows from [F4].

Source notes

Kinnunen proves Morrey's inequality by combining the ball oscillation estimate (Lemma 5.22, reproduced in the preceding item) with Holder's inequality in the form $\int_{B (x, r)} ∣ D u (w) ∣ ∣ y - w ∣^{1 - n} d w \leq c r^{1 - n / p} ∥ D u ∥_{L^{p} (B (x, r))}$ , printed pp. 140 and 77-79; the present proof follows that route and records the two-regime comparison of means needed because the statement normalizes the right-hand norm on the fixed ball $B (x, 2 r)$ . The continuity of the representative and the identification $u^{*} = u$ almost everywhere are the standard Lebesgue-point argument.

TheoremStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

$W^{1, \infty}$ functions on convex domains have Lipschitz representatives

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $n \geq 1$ and let $Ω \subseteq R^{n}$ be open, bounded and convex. If $u \in W^{1, \infty} (Ω; K)$ , then $u$ has a representative $u^{*}$ with $∣ u^{*} (x) - u^{*} (y) ∣ \leq ∥ D u ∥_{L^{\infty} (Ω)} ∣ x - y ∣ (x, y \in Ω),$ and $u^{*} = u$ almost everywhere on $Ω$ . Conversely, every Lipschitz function $f : Ω \to K$ with constant $L$ lies in $W^{1, \infty} (Ω)$ and $∥ D f ∥_{L^{\infty} (Ω)} \leq L$ for real scalars, while $∥ D f ∥_{L^{\infty} (Ω)} \leq \sqrt{2} L$ for complex scalars. In both cases the real-linear derivative has operator norm at most $L$ almost everywhere.

Gradient convention. For a weak gradient $D v = (D_{1} v, \dots, D_{n} v)$ we write $∣ D v ∣ = (\sum_{i = 1}^{n} ∣ D_{i} v ∣^{2})^{1 / 2}$ for its Euclidean norm and $∥ D v ∥_{L^{\infty} (Ω)} : = ∥ ∣ D v ∣ ∥_{L^{\infty} (Ω)}$ for the essential supremum of that norm, as in Pointwise potential bound for compactly supported smooth functions. For real scalars this norm equals the derivative operator norm. For complex scalars it is the Frobenius norm of the real-linear map $R^{n} \to R^{2}$ and can exceed its operator norm: $f (x) = x_{1} + i x_{2}$ is $1$ -Lipschitz but $∣ D f ∣ = \sqrt{2}$ . If $Ω = \emptyset$ both assertions hold vacuously.

Facts & Assumptions

Given: The Axiom of Choice; an integer $n \geq 1$ ; an open, bounded, convex set $Ω \subseteq R^{n}$ ; a field $K \in {R, C}$ ; for the first assertion a class $u \in W^{1, \infty} (Ω; K)$ and the finite number $L : = ∥ D u ∥_{L^{\infty} (Ω)}$ ; for the second assertion a function $f : Ω \to K$ that is Lipschitz with constant $L$ .

[F1]

$W^{k, p} (Ω; K)$ consists of the $L^{p}$ classes whose weak derivatives of order at most $k$ exist as $L^{p}$ classes; membership of $u$ in $W^{1, \infty}$ means that $u$ and all $D_{i} u$ lie in $L^{\infty}$ (Integer-order Sobolev spaces and their norms); an element of $L^{p}$ is an almost-everywhere class of measurable functions (The space $L^{p} (μ)$ as the quotient by null functions).

[F2]

$D_{i} u$ is the weak $\partial_{i}$ -derivative of $u$ exactly when $\int_{Ω} u \partial_{i} φ = - \int_{Ω} D_{i} u φ$ for every $φ \in C_{c}^{\infty} (Ω)$ , and weak derivatives are linear in the class (Weak derivative of a locally integrable function, Linearity, locality, and commutation of weak derivatives).

[F3]

The Axiom of Choice gives choice functions for arbitrary families of nonempty sets and implies Countable Choice and the prescribed-start form of Dependent Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, AC supplies the countable and dependent choices used in Banach integration).

[F4]

Every Euclidean ball has positive finite Lebesgue measure, measures are monotone under inclusion, and a bounded subset of $R^{n}$ has finite measure; in particular a nonempty open subset of $R^{n}$ contains a ball (Euclidean balls have positive finite Lebesgue measure, Measures are monotone, Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure).

[F5]

On a finite measure space every $L^{\infty}$ class lies in $L^{p}$ for every finite $p$ , with $∥ g ∥_{p} \leq μ (X)^{1 / p} ∥ g ∥_{\infty}$ (Finite-measure $L^{r}$ includes into $L^{p}$ for $p < r$ ).

[F6]

Under Countable Choice the interior mollifications $u_{ε} = ρ_{ε} * \tilde{u}$ of a representative of $u$ extended by zero are smooth on $R^{n}$ and satisfy $u_{ε} \to u$ in $W^{k, p} (U; K)$ for every open $U \subset \subset Ω$ , for $1 \leq p < \infty$ (Local smooth approximation in integer-order Sobolev spaces).

[F7]

$L^{p}$ is complete and every norm-convergent sequence in $L^{p}$ has an almost-everywhere convergent subsequence (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , Complex Lp completeness and almost-everywhere subsequences).

[F8]

The ball average $A_{r} f (x) = ∣ B (x, r) ∣^{- 1} \int_{B (x, r)} f$ is defined for $f \in L_{l o c}^{1} (R^{n})$ , and $A_{r} f (x) \to f (x)$ as $r ↓ 0$ for almost every $x$ (The average of a locally integrable function over a Euclidean ball, Lebesgue differentiation theorem on $R^{n}$ ).

[F9]

On completed sigma-finite products, nonnegative measurable functions may be integrated in either order (Tonelli-Fubini) (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability).

[F10]

For a $C^{1}$ diffeomorphism $T : U \to V$ and nonnegative Borel $h$ one has $\int_{V} h = \int_{U} h \circ T ∣ \det D T ∣$ (Borel change of variables from the compact-support formula and Radon uniqueness); an invertible linear map scales Lebesgue measure by $∣ \det ∣$ (A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not).

[F11]

If $v : [a, b] \to R^{m}$ is differentiable with integrable derivative then $\int_{a}^{b} v^{'} = v (b) - v (a)$ , and the chain rule computes the derivative of $t \mapsto u_{ε} (x + t (y - x))$ (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F12]

A Lipschitz function on a compact interval is absolutely continuous, and an absolutely continuous function is differentiable almost everywhere with derivative in $L^{1}$ and satisfies the fundamental theorem of calculus ( $C^{1}$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation, Fundamental theorem of calculus for absolutely continuous functions).

[F13]

If $g_{k} \to g$ almost everywhere and $∣ g_{k} ∣ \leq G$ almost everywhere for a single integrable $G$ , then $\int g_{k} \to \int g$ (Dominated convergence).

[F14]

A uniformly continuous map from a dense subset of a metric space into a complete metric space extends uniquely to a continuous map of the whole space (A uniformly continuous map from a dense subspace into a complete metric space extends uniquely to a uniformly continuous map on the whole space).

[F15]

Lipschitz with constant $L$ means $∣ f (x) - f (y) ∣ \leq L ∣ x - y ∣$ for all $x, y$ (Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction).

[F16]

Under Countable Choice, Lebesgue measure is the completion of Borel Lebesgue measure, and completion-measurable functions have almost-everywhere equal Borel representatives ( $L (R^{n})$ is exactly the completion of the restriction of $λ_{n}$ to the Borel sets, A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra); apply this to each real component.

Proof

technique · direct

1.1F1F2F4F5givenalgebra

Prepare the $W^{1, 1}$ class. If $Ω = \emptyset$ both assertions hold vacuously; assume $Ω \neq \emptyset$ . Then $0 < λ (Ω) < \infty$ by [F4]. Since $u \in W^{1, \infty} (Ω; K)$ , the class $u$ and every $D_{i} u$ lie in $L^{\infty}$ by [F1], and $∣ D u ∣ \leq L$ almost everywhere. By [F5] applied to the finite measure space $Ω$ , also $u \in L^{1} (Ω; K)$ and $D_{i} u \in L^{1} (Ω; K)$ . The test-function identities of [F2] that define the weak derivatives $D_{i} u$ are the same identities with the same test functions, so they remain valid with the $L^{1}$ in place of the $L^{\infty}$ classes: $u \in W^{1, 1} (Ω; K)$ , its $W^{1, 1}$ weak gradient is the given $D u$ , and $∣ D u ∣ \leq L$ almost everywhere.

1.2F3F6F7givenconstruct

Choose smooth approximations and a single subsequence. Let $u_{ε} = ρ_{ε} * \tilde{u}$ be the interior mollifications of [F6], where $\tilde{u}$ is the zero extension of a representative of $u$ ; then $u_{ε} \in C^{\infty} (R^{n}; K)$ and $u_{ε} \to u$ in $W^{1, 1} (U; K)$ for every open $U \subset \subset Ω$ . Put $U_{m} : = {x \in Ω : dist (x, R^{n} ∖ Ω) > 1 / m} \cap B (0, m)$ for $m \geq 1$ ; each $U_{m}$ is open and convex, $U_{m} \subseteq U_{m + 1} \subset \subset Ω$ , and $⋃_{m} U_{m} = Ω$ . Countable Choice, available from the Axiom of Choice by [F3], is what [F6] uses here. For every $m$ the family $(u_{ε})_{ε > 0}$ converges to $u$ in $W^{1, 1} (U_{m}; K)$ as $ε \to 0$ , so by [F7] we may extract recursively for $m = 1, 2, \dots$ a subsequence converging almost everywhere on $U_{m}$ ; the resulting diagonal sequence, rewritten as $ε_{k} ↓ 0$ , satisfies: $u_{ε_{k}} \to u$ almost everywhere on $Ω$ , and for every $m$ , $u_{ε_{k}} \to u$ in $W^{1, 1} (U_{m}; K)$ and $D u_{ε_{k}} \to D u$ in $L^{1} (U_{m}; K)$ . Each $u_{ε_{k}}$ is smooth on all of $R^{n}$ .

1.3F9F10algebra

A multiplicity estimate. Let $U \subseteq Ω$ be open convex with $λ (U) < \infty$ and let $g : R^{n} \to [0, \infty]$ be Borel measurable. Then $\int_{U} \int_{U} \int_{0}^{1} g ((1 - t) x + t y) d t d y d x \leq C (n) λ (U) \int_{R^{n}} g$ , where $C (n) : = 2 \int_{1 / 2}^{1} t^{- n} d t$ is finite and depends only on $n$ . Indeed, fix $t \in [1 / 2, 1]$ and $x \in U$ . The map $y \mapsto z = (1 - t) x + t y$ is an affine diffeomorphism of $R^{n}$ with linear part $t I$ and $∣ \det (t I) ∣ = t^{n}$ , so change of variables [F10] gives $\int_{U} g ((1 - t) x + t y) d y = t^{- n} \int_{(1 - t) x + t U} g (z) d z \leq t^{- n} \int_{R^{n}} g$ ; integrating over $x \in U$ and $t \in [1 / 2, 1]$ gives at most $(\int_{1 / 2}^{1} t^{- n} d t) λ (U) \int g$ . Symmetrically, for fixed $t \in [0, 1 / 2]$ and $y \in U$ the substitution $x \mapsto z = (1 - t) x + t y$ has linear part $(1 - t) I$ , so $\int_{U} g ((1 - t) x + t y) d x \leq (1 - t)^{- n} \int g$ ; integrating over $y \in U$ and $t \in [0, 1 / 2]$ and substituting $t \mapsto 1 - s$ gives the same bound. Tonelli's theorem [F9] justifies all iterated integrals, and adding the two halves gives the estimate.

1.4F5F9F12F15givenconstruct

The converse: $f$ is bounded and its a.e. partial derivatives are bounded functions. Assume $Ω \neq \emptyset$ and fix $x_{0} \in Ω$ . For every $x \in Ω$ the Lipschitz condition [F15] gives $∣ f (x) ∣ \leq ∣ f (x_{0}) ∣ + L ∣ x - x_{0} ∣ \leq ∣ f (x_{0}) ∣ + L diam (Ω) < \infty$ , so $f$ is bounded, hence $f \in L^{\infty} (Ω; K)$ and $f \in L^{1} (Ω; K)$ by [F5]. For $i \in {1, \dots, n}$ define $g_{i} (x)$ to be the limit of $(f (x + e_{i} / k) - f (x)) / (1 / k)$ where that limit exists, and $0$ elsewhere, the quotient being declared $0$ when $x + e_{i} / k \notin Ω$ . Each quotient is continuous on its open domain and has modulus at most $L$ there, so $g_{i}$ is measurable with $∣ g_{i} ∣ \leq L$ everywhere (use componentwise convergence for complex values) on $Ω$ . Moreover $g_{i} = \partial_{i} f$ almost everywhere: for each fixed value of the other $n - 1$ coordinates the section $t \mapsto f (\dots, t, \dots)$ is $L$ -Lipschitz on an interval, hence absolutely continuous and differentiable at almost every $t$ by [F12], and by Fubini [F9] the set of $x \in Ω$ at which the classical partial derivative fails to exist has measure zero.

2.1F7F11F16step 1.2step 1.3algebra

Segment identities and the integrated error. Each $u_{ε_{k}}$ is smooth on $R^{n}$ , so for all $x, y \in R^{n}$ the chain rule and the fundamental theorem of calculus [F11] give $u_{ε_{k}} (y) - u_{ε_{k}} (x) = \int_{0}^{1} D u_{ε_{k}} (x + t (y - x)) \cdot (y - x) d t$ . Choose finite-valued Borel representatives of the components of $D u$ by [F16]. Fix $m$ and apply step 1.3 with $U = U_{m}$ and $g = 1_{U_{m}} ∣ D u_{ε_{k}} - D u ∣$ , extended by zero: by convexity $x + t (y - x) \in U_{m}$ whenever $x, y \in U_{m}$ , so $\int_{U_{m}} \int_{U_{m}} \int_{0}^{1} ∣ D u_{ε_{k}} - D u ∣ (x + t (y - x)) d t d y d x \leq C (n) λ (U_{m}) ∥ D u_{ε_{k}} - D u ∥_{L^{1} (U_{m}; K)}$ , and the right-hand side tends to $0$ as $k \to \infty$ by step 1.2. Passing to a further subsequence, once more chosen diagonally over $m$ and relabelled, we may suppose that for every $m$ and almost every $(x, y) \in U_{m} \times U_{m}$ the integral $\int_{0}^{1} ∣ D u_{ε_{k}} - D u ∣ (x + t (y - x)) d t$ tends to $0$ .

2.2F2F10F13step 1.4givenalgebra

The converse: the functions $g_{i}$ are the weak derivatives and $f \in W^{1, \infty}$ . Let $φ \in C_{c}^{\infty} (Ω; K)$ . Fix $i$ and write $h = 1 / k$ . Since $φ$ is smooth with compact support, the difference quotients $(φ (x) - φ (x - h e_{i})) / h$ converge uniformly to $\partial_{i} φ$ as $h \to 0$ , and $f \in L^{1} (Ω; K)$ by step 1.4, so $\int_{Ω} f (x) (φ (x) - φ (x - h e_{i})) / h d x \to \int_{Ω} f \partial_{i} φ$ . Substituting $x = y + h e_{i}$ and using the translation case of the change-of-variables identity [F10] (determinant one) turns that integral into $\int_{Ω} (f (x) - f (x + h e_{i})) / h φ (x) d x$ . On the compact set $supp φ \subset \subset Ω$ and for $k$ large both $x$ and $x + e_{i} / k$ lie in $Ω$ , so there the quotient $q_{k} (x) : = (f (x + e_{i} / k) - f (x)) / (1 / k)$ is a Lipschitz difference quotient with $∣ q_{k} ∣ \leq L$ , and $q_{k} \to \partial_{i} f$ almost everywhere by step 1.4; dominated convergence [F13] with dominating function $L ∣ φ ∣$ on the finite measure set $supp φ$ therefore gives $\int_{Ω} f \partial_{i} φ = - \int_{Ω} g_{i} φ$ . By the defining identity [F2], $g_{i} = D_{i} f$ for every $i$ , and with $f \in L^{\infty}$ from step 1.4 and $g_{i} \in L^{\infty}$ this shows $f \in W^{1, \infty} (Ω; K)$ .

3.1F9step 1.1step 1.2step 1.3step 2.1givenalgebra

The almost-everywhere pair bound. Fix $m$ . By Fubini [F9], for almost every $(x, y) \in U_{m} \times U_{m}$ both $x$ and $y$ are points of almost-everywhere convergence of $u_{ε_{k}}$ toward $u$ and at the same time the integral convergence of step 2.1 holds. For such a pair define $u^{*} (x) : = \lim_{k} u_{ε_{k}} (x)$ and $u^{*} (y)$ similarly, and extend $u^{*}$ arbitrarily elsewhere. Passing to the limit in the identity of step 2.1, the left-hand side tends to $u^{*} (y) - u^{*} (x)$ , while the difference of the right-hand sides obeys $∣ \int_{0}^{1} [D u_{ε_{k}} - D u] (x + t (y - x)) \cdot (y - x) d t ∣ \leq ∣ y - x ∣ \int_{0}^{1} ∣ D u_{ε_{k}} - D u ∣ (x + t (y - x)) d t \to 0$ ; hence $u^{*} (y) - u^{*} (x) = \int_{0}^{1} D u (x + t (y - x)) \cdot (y - x) d t$ , where $∣ D u ∣ \leq L$ for almost every parameter on almost every pair segment: apply step 1.3 to the Borel null set where the chosen gradient exceeds $L$ , whose indicator has zero integral. Thus the integral is finite for these pairs. Therefore $∣ u^{*} (y) - u^{*} (x) ∣ \leq L ∣ y - x ∣$ . Since $Ω \times Ω = ⋃_{m} (U_{m} \times U_{m})$ and a countable union of null sets is null, there is one measurable representative $u^{*}$ of $u$ (the almost-everywhere limit of the subsequence of step 1.2) and one null set $N \subseteq Ω \times Ω$ such that $∣ u^{*} (x) - u^{*} (y) ∣ \leq L ∣ x - y ∣$ whenever $(x, y) \notin N$ .

3.2F2F9F10F12F13step 2.2givenalgebra

The converse: the derivative operator norm bound. Let $v$ be a unit vector. Choose $j$ with $v_{j} \neq 0$ and define the invertible linear map $T$ by $T e_{j} = v$ and $T e_{i} = e_{i}$ for $i \neq j$ ; then $∣ \det T ∣ = ∣ v_{j} ∣ > 0$ . The function $g : = f \circ T$ is Lipschitz on the open set $T^{- 1} Ω$ , and for each fixed value of the other coordinates its sections $t \mapsto g (\dots, t, \dots)$ are Lipschitz on intervals, hence differentiable at almost every $t$ by [F12]; Fubini [F9] shows that the set of $w \in T^{- 1} Ω$ at which the $e_{j}$ -directional derivative of $g$ fails to exist is null, and therefore, by the linear change-of-variables identity [F10], the set of $x \in Ω$ at which the $v$ -directional derivative of $f$ fails to exist is null. Define $q_{k} (x) : = (f (x + v / k) - f (x)) / (1 / k)$ when $x \in Ω$ and $x + v / k \in Ω$ , and $q_{k} (x) : = 0$ otherwise; then $q_{k}$ is measurable, $∣ q_{k} ∣ \leq L$ wherever the quotient is defined, and $q_{k} \to \partial_{v} f$ almost everywhere on $Ω$ . For $φ \in C_{c}^{\infty} (Ω; K)$ the same difference-quotient computation as in step 2.2, now in the direction $v$ , gives $\int_{Ω} f \partial_{v} φ = - \lim_{k} \int_{Ω} q_{k} φ = - \int_{Ω} h_{v} φ$ , where $h_{v}$ is the limit of $q_{k}$ where it exists, and $0$ elsewhere satisfies $∣ h_{v} ∣ \leq L$ everywhere and equals $\partial_{v} f$ almost everywhere. Hence $h_{v}$ is a representative of the weak directional derivative $D_{v} f : = \sum_{i} v_{i} D_{i} f$ of [F2], and $∣ \sum_{i} v_{i} D_{i} f ∣ \leq L$ almost everywhere for every unit vector $v$ . Now let $V$ be a countable dense subset of the unit sphere. Choosing representatives of $D_{1} f, \dots, D_{n} f$ , for each $v \in V$ the identity $\sum_{i} v_{i} D_{i} f = h_{v}$ holds almost everywhere, so on a set of full measure all the countably many inequalities $∣ \sum_{i} v_{i} D_{i} f ∣ \leq L$ hold simultaneously. At every such $x$ one has $∥ D f (x) ∥_{o p} = \sup_{∣ w ∣ = 1} ∣ \sum_{i} w_{i} D_{i} f (x) ∣ = \sup_{v \in V} ∣ \sum_{i} v_{i} D_{i} f (x) ∣ \leq L$ , because $w \mapsto \sum_{i} w_{i} D_{i} f (x)$ is continuous and $V$ is dense in the unit sphere. For real $f$ this operator norm is $∣ D f ∣$ . For $f = a + i b$ , each row gradient satisfies $∣ D a ∣, ∣ D b ∣ \leq L$ , so $∣ D f ∣^{2} = ∣ D a ∣^{2} + ∣ D b ∣^{2} \leq 2 L^{2}$ . This gives the asserted real and complex Euclidean bounds and completes the converse.

4.1F8step 3.1algebra

Lebesgue points and the doubling comparison. Let $A_{r} f$ be the ball average of [F8], applied to the zero extension of a representative of $u$ ; by [F8] the set $E$ of $x \in Ω$ with $A_{r} u (x) \to u (x)$ as $r ↓ 0$ has full measure in $Ω$ , and on $E$ the limit equals $u^{*} (x)$ because $u^{*} = u$ almost everywhere. Replace $E$ by its intersection with the full-measure set where $u^{*} = u$ . For $x, y \in E$ and $0 < r < \min (dist (x, \partial Ω), dist (y, \partial Ω))$ one has $A_{r} u (x) - A_{r} u (y) = ∣ B (x, r) ∣^{- 1} ∣ B (y, r) ∣^{- 1} \int_{B (x, r)} \int_{B (y, r)} (u^{*} (z) - u^{*} (z^{'})) d z^{'} d z$ , so using that the exceptional set $N$ of step 3.1 is null, $∣ A_{r} u (x) - A_{r} u (y) ∣ \leq L ∣ B (x, r) ∣^{- 1} ∣ B (y, r) ∣^{- 1} \int_{B (x, r)} \int_{B (y, r)} ∣ z - z^{'} ∣ d z^{'} d z \leq L (∣ x - y ∣ + 2 r)$ , because $∣ z - z^{'} ∣ \leq ∣ z - x ∣ + ∣ x - y ∣ + ∣ y - z^{'} ∣ \leq 2 r + ∣ x - y ∣$ on the two balls. Letting $r ↓ 0$ yields $∣ u^{*} (x) - u^{*} (y) ∣ \leq L ∣ x - y ∣$ for all $x, y \in E$ .

5.1F4F14step 4.1givenalgebra∎

Extending to a Lipschitz representative and completing the first assertion. The full-measure set $E$ is dense in $Ω$ : otherwise $E$ would be disjoint from some ball $B \subseteq Ω$ , and $λ (B) > 0$ by [F4], contradicting $λ (Ω ∖ E) = 0$ . On $E$ the restriction of $u^{*}$ is $L$ -Lipschitz, hence uniformly continuous. By [F14] applied with $X : = \overline{Ω}$ , the dense subset $E$ and the complete target $K$ , there is a continuous $g : \overline{Ω} \to K$ with $g ∣_{E} = u^{*} ∣_{E}$ . For arbitrary $x, y \in \overline{Ω}$ choose $x_{j}, y_{j} \in E$ with $x_{j} \to x$ and $y_{j} \to y$ ; by continuity $g (x) = \lim_{j} g (x_{j})$ and $g (y) = \lim_{j} g (y_{j})$ , so step 4.1 gives $∣ g (x) - g (y) ∣ = \lim_{j} ∣ u^{*} (x_{j}) - u^{*} (y_{j}) ∣ \leq L \lim_{j} ∣ x_{j} - y_{j} ∣ = L ∣ x - y ∣$ . Thus $g ∣_{Ω}$ is an $L$ -Lipschitz representative of $u$ on $Ω$ that equals $u$ almost everywhere, and the first assertion of the theorem is proved.

Source notes

Kinnunen proves Theorem 3.31 in $R^{n}$ by reducing to $W_{l o c}^{1, p}$ for $p > n$ , importing the $p > n$ Sobolev embedding and finishing by mollification with uniform convergence. That route is not available at this position in the reading order, where the $p > n$ theory is still to come, so the proof above uses only the density of smooth functions, Fubini, the one-dimensional theory of absolutely continuous functions and Lebesgue's differentiation theorem. The key substitute for the $p > n$ embedding is the multiplicity estimate 1.3, which lets the classical segment identities for smooth approximations pass to the limit for almost every pair of points; the sharp constant is then obtained from ball averages and the dense-subset extension theorem. The converse direction identifies the weak gradient of a Lipschitz function by difference quotients rather than by Rademacher's theorem; the $v$ -directional derivative is transferred to a coordinate direction by an explicit invertible linear map, whose only measure-theoretic input is the linear change-of-variables identity for Lebesgue measure. Hunter's notes and Teschl's chapter cover the Lipschitz/absolute-continuity interface used in the converse; no result of those sources is used beyond that interface.

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Weak partial derivatives lower the Sobolev order

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $Ω \subseteq R^{n}$ be open, $k \geq 1$ , $1 \leq p \leq \infty$ and $u \in W^{k, p} (Ω; K)$ . For every multi-index $β$ with $∣ β ∣ \leq k$ the weak derivative $D^{β} u$ , viewed as an $L^{p}$ class, belongs to $W^{k - ∣ β ∣, p} (Ω)$ , and $D^{α} (D^{β} u) = D^{α + β} u$ a.e. whenever $∣ α ∣ + ∣ β ∣ \leq k$ .

Facts & Assumptions

Given: Countable Choice; an open $Ω \subseteq R^{n}$ with $n \geq 1$ ; integers $k \geq 1$ ; an exponent $1 \leq p \leq \infty$ ; a class $u \in W^{k, p} (Ω; K)$ ; and multi-indices $β, α$ with $∣ β ∣ \leq k$ and $∣ α ∣ + ∣ β ∣ \leq k$ .

[F1]

$u \in W^{k, p} (Ω; K)$ means that $u$ and all weak derivatives $D^{γ} u$ with $∣ γ ∣ \leq k$ have representatives in $L^{p} (Ω; K)$ (Integer-order Sobolev spaces and their norms).

[F2]

A weak derivative is defined by the test-function integration-by-parts identity against $C_{c}^{\infty} (Ω)$ test functions (Weak derivative of a locally integrable function).

[F3]

If $D^{α} u$ and $D^{β} u$ exist in $L_{l o c}^{1} (Ω)$ and $D^{α + β} u$ also exists in $L_{l o c}^{1} (Ω)$ , then $D^{α} (D^{β} u)$ exists and equals $D^{α + β} u$ almost everywhere (Linearity, locality, and commutation of weak derivatives).

[F4]

Weak derivatives in $L_{l o c}^{1}$ are unique as almost-everywhere classes (Uniqueness of a weak derivative as an almost-everywhere class).

[F5]

Countable Choice, assumed throughout (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1F1F2F5given

Regularity bookkeeping. Since $u \in W^{k, p} (Ω; K)$ and $∣ β ∣, ∣ α + β ∣ \leq k$ , the classes $D^{α} u$ , $D^{β} u$ and $D^{α + β} u$ are defined and lie in $L^{p} (Ω; K)$ by [F1]; representatives of $L^{p}$ classes are locally integrable on the open set $Ω$ , so both are classes in $L_{l o c}^{1} (Ω; K)$ to which the weak-differentiation calculus of [F2] applies.

2.1F3step 1.1given

Commutation. With $α, β$ as in the Given, the hypotheses of [F3] are met by step 1.1: $D^{α} u$ , $D^{β} u$ and $D^{α + β} u$ exist in $L_{l o c}^{1} (Ω)$ , hence $D^{α} (D^{β} u)$ exists and satisfies $D^{α} (D^{β} u) = D^{α + β} u$ almost everywhere on $Ω$ .

3.1F1F2F4step 1.1step 2.1algebra∎

Descent of the order. By step 2.1, for every multi-index $α$ with $∣ α ∣ \leq k - ∣ β ∣$ the weak derivative $D^{α} (D^{β} u)$ exists and equals $D^{α + β} u$ , which lies in $L^{p} (Ω; K)$ by step 1.1; also $D^{β} u \in L^{p} (Ω; K)$ . By [F1] this says exactly that the class $D^{β} u$ belongs to $W^{k - ∣ β ∣, p} (Ω; K)$ . The identity $D^{α} (D^{β} u) = D^{α + β} u$ is an almost-everywhere identity of classes, and by uniqueness [F4] it is independent of the representatives chosen for $D^{β} u$ and $D^{α + β} u$ .

Source notes

The source records the commutation of weak partial derivatives as part of the elementary calculus of weak derivatives; the proof above isolates the two uses: existence of the higher derivative and uniqueness of the $L_{l o c}^{1}$ classes. No regularity of $\partial Ω$ and no boundedness of $Ω$ is needed.

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Higher-order Sobolev embedding

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $n \geq 2$ , let $Ω$ be a bounded $W^{k, p}$ -extension domain, $k \geq 1$ and $1 \leq p < \infty$ . Let $K \in {R, C}$ and $u \in W^{k, p} (Ω; K)$ . Then:

if $k p < n$ , $∥ u ∥_{L^{q} (Ω)} \leq C ∥ u ∥_{W^{k, p} (Ω)}$ for every $q$ with $\frac{1}{q} \geq \frac{1}{p} - \frac{k}{n}$ (equivalently $1 \leq q \leq \frac{n p}{n - k p}$ );
if $k p = n$ , $∥ u ∥_{L^{q} (Ω)} \leq C (q) ∥ u ∥_{W^{k, p} (Ω)}$ for every finite $q$ ;
if $k p > n$ , then for every integer $m \geq 0$ and every $0 < α < 1$ with $m + α < k - \frac{n}{p}$ there is a representative in $C^{m, α} (\overline{Ω})$ with $∥ u ∥_{C^{m, α} (\overline{Ω})} \leq C ∥ u ∥_{W^{k, p} (Ω)}$ ; when $k - \frac{n}{p} \notin Z$ one may take $m = ⌊ k - \frac{n}{p} ⌋$ and $α = k - \frac{n}{p} - m$ .

Here $C^{m, α} (\overline{Ω})$ uses the continuous derivatives of the constructed representative on an ambient neighbourhood of $\overline{Ω}$ , with norm $∥ g ∥_{C^{m, α} (\overline{Ω})} : = \sum_{∣ β ∣ \leq m} \sup_{\overline{Ω}} ∣ D^{β} g ∣ + \sum_{∣ β ∣ = m} \sup_{\begin{array}{c} x, y \in \overline{Ω} \\ x \neq y \end{array}} \frac{∣ D^{β} g (x) - D^{β} g (y) ∣}{∣ x - y ∣^{α}} .$ For $Ω = \emptyset$ set these norm values to $0$ . The proof supplies such an ambient representative, so no boundary differentiability or regularity of $\partial Ω$ is assumed.

Facts & Assumptions

Given: The Axiom of Choice; $n \geq 2$ ; a bounded extension domain $Ω$ ; $k \geq 1$ ; $1 \leq p < \infty$ ; a class $u \in W^{k, p} (Ω; K)$ .

[F1]

Lower-order derivatives: for every multi-index $β$ with $∣ β ∣ \leq k$ , $D^{β} u \in W^{k - ∣ β ∣, p} (Ω)$ and $∥ D^{β} u ∥_{W^{k - ∣ β ∣, p}} \leq C ∥ u ∥_{W^{k, p}}$ (Weak partial derivatives lower the Sobolev order, Integer-order Sobolev spaces and their norms, The notation $H^{k}$ and the reserved zero-boundary symbol, The space $L^{p} (μ)$ as the quotient by null functions).

[F2]

For $1 \leq q < n$ , the Sobolev conjugate $q^{*} = n q / (n - q)$ is finite and satisfies $1 / q^{*} = 1 / q - 1 / n$ ; iterating this relation gives $p_{k}^{*} = n p / (n - k p)$ when $k p < n$ (The Sobolev conjugate exponent and the scaling identity).

[F3]

The local Morrey estimate: for $n < q < \infty$ , $w \in W^{1, q}$ has a continuous representative and $[w]_{C^{0, 1 - n / q} (B (a, r))} \leq C (n, q) ∥ D w ∥_{L^{q} (B (a, 2 r))}$ when the doubled ball is compactly contained in its domain (Morrey's inequality for $p > n$ , Local Hölder and scaled C-two-alpha norms on balls).

[F4]

Holder's inequality on a bounded measurable set $B$ compares $L^{r} (B)$ and $L^{s} (B)$ for $1 \leq r \leq s < \infty$ (Holder's inequality for integrals, including the endpoint cases).

[F5]

The extension operator and the whole-space results are used through the extension domain hypothesis (Sobolev extension domains and extension operators); the Axiom of Choice is inherited from the suppliers.

[F6]

Because $Ω$ is bounded, $\overline{Ω}$ is compact. Choose a ball $B$ containing $\overline{Ω}$ , a smooth cutoff $η \in C_{c}^{\infty} (B)$ equal to $1$ on a neighbourhood of $\overline{Ω}$ , and a bounded extension operator $E : W^{k, p} (Ω) \to W^{k, p} (R^{n})$ . Then $v : = η E u$ is compactly supported in $B$ , belongs to $W^{k, p} (R^{n})$ , satisfies $v = u$ on $Ω$ , and $∥ v ∥_{W^{k, p} (R^{n})} \leq C ∥ u ∥_{W^{k, p} (Ω)}$ ; its weak derivatives restrict to those of $u$ on $Ω$ (Sobolev extension domains and extension operators, A Euclidean bump for a compact set inside an open set, Weak Leibniz rule with a smooth factor).

[F7]

The whole-space first-order inequality holds for $1 < q < n$ (The Gagliardo-Nirenberg-Sobolev inequality for $1 < p < n$ ). At $q = 1$ it extends to $W^{1, 1}$ from The p=1 Gagliardo-Nirenberg-Sobolev inequality using Compactly supported smooth functions are dense in W^{k,p}(R^n): apply the smooth estimate to differences; use Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , Complex Lp completeness and almost-everywhere subsequences for the $L^{n / (n - 1)}$ limit and successive almost-everywhere subsequences in that space and $L^{1}$ to identify it with the original class.

[F8]

For $q > n$ , apply the local Morrey estimate [F3] on a ball containing the compact support of $v$ ; it bounds the Holder seminorm of a representative by $C ∥ D v ∥_{L^{q}}$ ; its supremum on that ball is bounded by the average, at most $∣ B ∣^{- 1 / q} ∥ v ∥_{L^{q} (B)}$ , plus the oscillation bound. Hence it gives a $C^{0, 1 - n / q}$ representative with norm bounded by $C ∥ v ∥_{W^{1, q} (R^{n})}$ , in particular on $\overline{Ω}$ (Morrey's inequality for $p > n$ , Local Hölder and scaled C-two-alpha norms on balls).

[F9]

Continuous weak first derivatives are classical derivatives. On an inner ball, convolution of a continuous function converges uniformly on smaller compact balls, because $∣ (ρ_{ε} * f) (x) - f (x) ∣ \leq \sup_{∣ h ∣ \leq ε} ∣ f (x - h) - f (x) ∣$ and continuity on the compact neighbourhood is uniform. The same applies to its continuous weak derivatives; Interior mollification commutes with weak derivatives identifies their convolutions with derivatives of the smooth mollification. Pass to the limit in the coordinate segment identity from If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz to obtain $f (x + t e_{i}) - f (x) = \int_{0}^{t} g_{i} (x + s e_{i}) d s$ . Differentiating this identity gives $\partial_{i} f = g_{i}$ ; repeat for higher orders. Weak representatives are unique (Uniqueness of a weak derivative as an almost-everywhere class).

Proof

1.1F1F2F5F6F7algebra

Whole-space iteration. Let $B$ be the fixed bounded support ball from [F6]. Suppose $1 \leq q_{0} < n$ and $w \in W^{ℓ, q_{0}} (R^{n})$ is supported in $B$ , with $ℓ \geq 1$ , and put $1 / q_{1} = 1 / q_{0} - 1 / n$ , so $q_{1} = q_{0}^{*}$ . For every $∣ β ∣ \leq ℓ - 1$ , [F1] gives $D^{β} w \in W^{1, q_{0}} (R^{n})$ with norm bounded by $C ∥ w ∥_{W^{ℓ, q_{0}}}$ . Applying the whole-space inequality [F7] to each derivative gives $D^{β} w \in L^{q_{1}}$ ; summing the finitely many norms shows $w \in W^{ℓ - 1, q_{1}} (R^{n})$ with controlled norm. Compact support is retained, and [F2] gives the invariant $ℓ - n / q_{0} = (ℓ - 1) - n / q_{1}$ . The $q_{0} = 1$ case uses the p=1 inequality and its density passage in [F7].

1.2F1F3F6F7F8algebra

Assertion (3): global Holder representatives. Assume $k p > n$ and choose $m \geq 0$ , $0 < α < 1$ with $m + α \leq k - \frac{n}{p}$ ; for the endpoint equality, require $k - \frac{n}{p} \notin Z$ and $m = ⌊ k - \frac{n}{p} ⌋$ . Fix $β$ with $∣ β ∣ \leq m$ and put $ℓ : = k - ∣ β ∣$ and $t : = n / p$ , so $ℓ - t \geq α$ . By [F1] and [F6], $w : = D^{β} v$ is compactly supported in $W^{ℓ, p} (R^{n})$ with norm at most $C ∥ u ∥_{W^{k, p} (Ω)}$ . Starting from $(ℓ, p)$ , whenever the current exponent $q < n$ and current order $r \geq 2$ , apply [F7] to every derivative of $w$ of order at most $r - 1$ ; this gives $w \in W^{r - 1, q^{*}}$ , where $1 / q^{*} = 1 / q - 1 / n$ , with controlled norm. The invariant $r - n / q = ℓ - n / p \geq α > 0$ shows the process cannot stop with order $1$ and exponent below $n$ . If it reaches $q > n$ , write the remaining order as $r$ . When $r = 1$ , take $q^{'} = q$ ; Morrey gives exponent $1 - n / q = ℓ - n / p \geq α$ . When $r \geq 2$ , the first derivatives of $w$ lie in $W^{1, q}$ ; [F8] bounds them and $w$ on the fixed compact support, so $w \in W^{1, q^{'}} (R^{n})$ for every finite $q^{'}$ , and choose $q^{'} > n$ with $1 - n / q^{'} > α$ . If the iteration reaches $q = n$ , then the invariant gives $r - n / q = r - 1 \geq α > 0$ , hence $r \geq 2$ . Thus $w$ and its first derivatives are compactly supported in $W^{1, n}$ ; on their common bounded support, Holder puts them in $W^{1, s}$ for any $1 < s < n$ sufficiently close to $n$ , and [F7] then puts them in $L^{s^{*}}$ for $s^{*} = n s / (n - s)$ , which can be chosen arbitrarily large. Hence again $w \in W^{1, q^{'}}$ for some $q^{'} > n$ with $1 - n / q^{'} > α$ . In every case $w \in W^{1, q^{'}}$ for an exponent $q^{'} > n$ with $1 - n / q^{'} \geq α$ , and its norm is bounded by $C ∥ u ∥_{W^{k, p}}$ .

2.1F2F4F6F7step 1.1algebra

Assertion (1). Assume $k p < n$ and take the compactly supported extension $v$ from [F6]. Iterating step 1.1 for $j = 1, \dots, k - 1$ gives $v \in W^{1, p_{k - 1}} (R^{n})$ with $1 / p_{j} = 1 / p - j / n$ ; all exponents are finite and $p_{j} < n$ because $(j + 1) p < n$ for $j \leq k - 1$ . One more application of [F7] gives $v \in L^{p_{k}^{*}} (R^{n})$ , where $1 / p_{k}^{*} = 1 / p - k / n$ by [F2]. For every $1 \leq q \leq p_{k}^{*}$ , Holder [F4] on the fixed support ball $B$ gives $∥ v ∥_{L^{q} (R^{n})} \leq C (B, p, q) ∥ v ∥_{L^{p_{k}^{*}} (R^{n})}$ ; the whole-space endpoint estimate and [F6] bound this by $C (n, k, p, q, Ω) ∥ u ∥_{W^{k, p} (Ω)}$ . Since $v = u$ on $Ω$ , restriction proves (1).

2.2F4F6F7step 1.1algebra

Assertion (2). Assume $k p = n$ , and take $v$ from [F6]. Iterating step 1.1 through $k - 1$ reductions gives $v \in W^{1, n} (R^{n})$ with support in $B$ . If $1 \leq q \leq n$ , Holder [F4] on $B$ bounds $∥ v ∥_{L^{q}}$ by $C (B, q) ∥ v ∥_{L^{n}}$ . If $q > n$ , set $s = n q / (n + q)$ , so $1 < s < n$ and $s^{*} = q$ ; compact support and Holder give $v \in W^{1, s} (R^{n})$ with $∥ v ∥_{W^{1, s}} \leq C (B, s) ∥ v ∥_{W^{1, n}}$ , and [F7] gives $∥ v ∥_{L^{q}} \leq C (n, s) ∥ v ∥_{W^{1, s}}$ . In both cases the norm is bounded by $C (n, k, q, Ω) ∥ u ∥_{W^{k, p} (Ω)}$ using [F6]; restriction to $Ω$ proves (2), with no $L^{\infty}$ endpoint asserted.

3.1F3F8F9step 1.2algebra∎

Apply [F3] and [F8] with this exponent $q^{'}$ to $w$ on a ball containing $\overline{Ω}$ . This gives a representative $g_{β} \in C^{0, α} (\overline{Ω})$ with $∥ g_{β} ∥_{C^{0, α} (\overline{Ω})} \leq C ∥ u ∥_{W^{k, p} (Ω)}$ , uniformly over the finitely many $∣ β ∣ \leq m$ . By [F9], whenever $∣ β ∣ < m$ , the classical derivatives of $g_{β}$ are the continuous representatives $g_{β + e_{i}}$ , since these represent the weak derivatives of $D^{β} v$ and weak derivatives are unique. Therefore $g_{0} \in C^{m, α} (\overline{Ω})$ , represents $u$ , and its $C^{m, α}$ norm is bounded by the sum of the finitely many bounds just obtained. This proves (3) for the strict range and also the stated fractional-endpoint case: there $ℓ - t = α$ for $∣ β ∣ = m$ , and the final Morrey exponent is exactly $α$ , which is allowed by [F8].

Source notes

The higher-order embedding is the iteration of the first-order Sobolev inequalities motivated by Kinnunen (Theorem 3.23 and the local higher-order iteration of Remark 3.41) and Teschl (Theorem 9.22), followed by Morrey's estimate. The compactly supported whole-space extension reduces the boundary claim to one fixed ball containing the domain closure; it is essential here because the local Morrey statement alone only controls balls compactly contained in the open set. The finite iteration stops at a supercritical exponent, or at the critical exponent with at least two derivatives remaining, and supplies the global closure-wide Holder norm claimed above.

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Weak product rule for bounded Sobolev functions

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $Ω \subseteq R^{n}$ be open, $1 \leq p < \infty$ , and let $u, v \in W^{1, p} (Ω; K) \cap L^{\infty} (Ω)$ . Then $u v \in W^{1, p} (Ω)$ and $D_{j} (u v) = (D_{j} u) v + u (D_{j} v)$ a.e. for every $j$ .

Facts & Assumptions

Given: The Axiom of Choice; an open $Ω \subseteq R^{n}$ with $n \geq 1$ ; an exponent $1 \leq p < \infty$ ; and classes $u, v \in W^{1, p} (Ω; K) \cap L^{\infty} (Ω)$ .

[F1]

$W^{1, p} (Ω; K)$ consists of the $L^{p}$ classes whose weak first derivatives exist as $L^{p}$ classes (Integer-order Sobolev spaces and their norms), and $L^{p}$ is the quotient by almost-everywhere null functions (The space $L^{p} (μ)$ as the quotient by null functions).

[F2]

Chain rule for a globally Lipschitz scalar function $F$ : for real $w \in W^{1, p} (Ω; R)$ , $F \circ w \in W^{1, p}$ whenever $F (0) = 0$ , and $D_{i} (F \circ w)$ agrees almost everywhere with $F^{'} (w) D_{i} w$ where $F$ is differentiable at $w$ , with the product defined as $0$ on the preimage of the nondifferentiability set, which need not itself be null (Chain rule for globally Lipschitz scalar maps of Sobolev functions).

[F3]

Weak differentiation is linear and local: weak derivatives of linear combinations are the corresponding linear combinations, and they restrict to open subsets (Linearity, locality, and commutation of weak derivatives).

[F5]

The Axiom of Choice, used through the chain-rule interface of [F2] (The Axiom of Choice).

Proof

technique · direct

1.1F1givenalgebra

Integrability of the products. Since $v \in L^{\infty} (Ω)$ and $u \in L^{p} (Ω)$ , the pointwise bound $∣ u v ∣^{p} \leq ∥ v ∥_{\infty}^{p} ∣ u ∣^{p}$ , followed by integration, gives $∥ u v ∥_{L^{p}} \leq ∥ u ∥_{L^{p}} ∥ v ∥_{L^{\infty}} < \infty$ , and the same argument applies to $u (D_{j} v)$ and $(D_{j} u) v$ because $D_{j} u, D_{j} v \in L^{p}$ and $u, v \in L^{\infty}$ . Thus the three classes $u v$ , $(D_{j} u) v$ and $u (D_{j} v)$ all lie in $L^{p} (Ω; K)$ , and so does their sum $(D_{j} u) v + u (D_{j} v)$ (taken componentwise for $K = C$ ).

1.2F2F5givenalgebra

The real case for a truncated square. Let $w \in W^{1, p} (Ω; R) \cap L^{\infty} (Ω)$ and $M : = ∥ w ∥_{L^{\infty}}$ . Define $G_{M} (t) : = t^{2}$ for $∣ t ∣ \leq M$ and $G_{M} (t) : = 2 M ∣ t ∣ - M^{2}$ for $∣ t ∣ > M$ . Then $G_{M}$ is in $C^{1} (R)$ , with $G_{M}^{'} (t) = 2 t$ for $∣ t ∣ \leq M$ and $G_{M}^{'} (t) = 2 M sgn (t)$ for $∣ t ∣ > M$ , it is globally Lipschitz with constant $2 M$ , $G_{M} (0) = 0$ , and $G_{M} (t) = t^{2}$ for $∣ t ∣ \leq M$ . Since $∣ w ∣ \leq M$ almost everywhere, $G_{M} \circ w = w^{2}$ almost everywhere and $G_{M}^{'} (w) = 2 w$ almost everywhere on $Ω$ ; by [F2] the class $G_{M} \circ w$ lies in $W^{1, p} (Ω)$ with $D_{j} (G_{M} \circ w) = G_{M}^{'} (w) D_{j} w = 2 w D_{j} w$ almost everywhere. In particular $w^{2} = G_{M} \circ w \in W^{1, p} (Ω)$ and $D_{j} (w^{2}) = 2 w D_{j} w$ .

2.1F3step 1.1step 1.2algebra

Polarization in the real case. Suppose first that $K = R$ , and put $w_{\pm} : = u \pm v \in W^{1, p} (Ω; R) \cap L^{\infty} (Ω)$ ; these classes lie in $W^{1, p}$ by the linearity part of [F3]. Applying $G_{M_{\pm}}$ of step 1.2 with $M_{\pm} : = ∥ w_{\pm} ∥_{L^{\infty}}$ gives $u v = \frac{1}{4} (w_{+}^{2} - w_{-}^{2})$ as $L^{p}$ classes and, by linearity of weak derivatives [F3], $D_{j} (u v) = \frac{1}{4} (D_{j} (w_{+}^{2}) - D_{j} (w_{-}^{2})) = \frac{1}{2} (w_{+} D_{j} w_{+} - w_{-} D_{j} w_{-}) = (D_{j} u) v + u (D_{j} v)$ almost everywhere.

3.1F1F3step 2.1algebra∎

Complex case and conclusion. For general $K \in {R, C}$ , write $u = u_{1} + i u_{2}$ and $v = v_{1} + i v_{2}$ with real components; these components lie in $W^{1, p} (Ω; R) \cap L^{\infty} (Ω)$ and $D_{j} u = D_{j} u_{1} + i D_{j} u_{2}$ , $D_{j} v = D_{j} v_{1} + i D_{j} v_{2}$ by the componentwise definition of the weak derivative [F1]. Applying step 2.1 to the four real products and using linearity [F3], $D_{j} (u v) = D_{j} ((u_{1} v_{1} - u_{2} v_{2}) + i (u_{1} v_{2} + u_{2} v_{1})) = (D_{j} u) v + u (D_{j} v)$ almost everywhere, and $u v \in W^{1, p} (Ω; K)$ by step 1.1.

Source notes

The classical route approximates $u$ and $v$ by smooth functions and passes to the limit in a closed graph; the proof above instead polarises the product and applies the published chain rule for globally Lipschitz scalar functions to a truncated square, which is available for all $1 \leq p < \infty$ and avoids any global smooth-approximation theorem. The boundedness of $u$ and $v$ is used through the truncation radius $M$ and in the integrability step.

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The Sobolev space $W^{k, p}$ is an algebra above the critical index

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $n \geq 2$ , let $Ω$ be a bounded $W^{k, p}$ -extension domain, $k \geq 1$ and $1 \leq p < \infty$ with $k p > n$ . Then there is $C (n, p, k, Ω)$ with $∥ u v ∥_{W^{k, p} (Ω)} \leq C ∥ u ∥_{W^{k, p} (Ω)} ∥ v ∥_{W^{k, p} (Ω)} (u, v \in W^{k, p} (Ω)),$ so $W^{k, p} (Ω)$ is a Banach algebra; the constants and the conclusion may depend on the choice of equivalent Sobolev norm only through $C$ .

Facts & Assumptions

Given: The Axiom of Choice; $n \geq 2$ ; a bounded extension domain $Ω$ ; $k \geq 1$ ; $1 \leq p < \infty$ with $k p > n$ ; and classes $u, v \in W^{k, p} (Ω; K)$ .

[F1]

For $w \in W^{k, p} (Ω)$ fix its bounded whole-space extension $E w$ and a ball $B \supset \overline{Ω}$ . By Weak partial derivatives lower the Sobolev order, $D^{γ} (E w) ∣_{B} \in W^{s, p} (B)$ with $s = k - ∣ γ ∣$ and norm at most $C ∥ w ∥_{W^{k, p} (Ω)}$ . A ball is a $W^{s, p}$ -extension domain for every integer $s \geq 1$ by Bounded C^k domains admit integer-order Sobolev extension. Apply Higher-order Sobolev embedding on $B$ and restrict to $Ω$ : $D^{γ} w$ is bounded if $s p > n$ , lies in every finite $L^{q}$ if $s p = n$ , and lies in $L^{q}$ for $1 / q \geq 1 / p - s / n$ if $0 < s p < n$ . If $s = 0$ , use its original $L^{p}$ bound. All norms are controlled by $C ∥ w ∥_{W^{k, p}}$ ; this needs only the fixed $W^{k, p}$ extension of $w$ , not extension operators for lower-order classes on $Ω$ (Sobolev extension domains and extension operators, Integer-order Sobolev spaces and their norms).

[F2]

Meyers-Serrin density: for $1 \leq p < \infty$ the smooth functions in $C^{\infty} (Ω) \cap W^{k, p} (Ω)$ are dense in $W^{k, p} (Ω)$ (Meyers–Serrin density on an arbitrary open set, whose Countable-Choice hypothesis is supplied by the Axiom of Choice assumed here; The space $L^{p} (μ)$ as the quotient by null functions).

[F3]

Holder's inequality in its multi-factor form: for nonnegative measurable $f, g$ with $\frac{1}{q_{1}} + \frac{1}{q_{2}} \leq \frac{1}{p}$ one has $∥ f g ∥_{L^{p} (Ω)} \leq C (Ω) ∥ f ∥_{L^{q_{1}}} ∥ g ∥_{L^{q_{2}}}$ on the finite measure domain: apply Holder to $∣ f ∣^{p}$ , $∣ g ∣^{p}$ and $1$ with reciprocal exponents $p / q_{1}$ , $p / q_{2}$ and $1 - p / q_{1} - p / q_{2}$ (iterate the two-factor inequality; an exponent $0$ means an $L^{\infty}$ factor). Taking $p$ -th roots gives the displayed bound (Holder's inequality for integrals, including the endpoint cases).

[F4]

The weak derivative is characterized by the test-function identity: an $L^{p}$ class $g_{α}$ is the weak $α$ -derivative of $w$ exactly when $\int_{Ω} w \partial^{α} φ = (- 1)^{∣ α ∣} \int_{Ω} g_{α} φ$ for every $φ \in C_{c}^{\infty} (Ω)$ (Weak derivative of a locally integrable function, Integer-order Sobolev spaces and their norms). For smooth functions the classical derivatives are the weak derivatives (Classical derivatives agree with weak derivatives).

[F5]

$W^{k, p} (Ω; K)$ is complete (Integer-order Sobolev spaces are Banach).

Proof

technique · direct

1.1F1F3givenalgebra

The product estimate for derivatives. Let $w_{1}, w_{2} \in W^{k, p} (Ω; K)$ and let $γ_{1}, γ_{2}$ be multi-indices with $∣ γ_{1} ∣ + ∣ γ_{2} ∣ \leq k$ ; put $s_{i} : = k - ∣ γ_{i} ∣ \geq 0$ , so that $s_{1} + s_{2} = 2 k - ∣ γ_{1} ∣ - ∣ γ_{2} ∣ \geq k$ . By [F1] choose exponents $q_{i} \in [1, \infty]$ with $D^{γ_{i}} w_{i} \in L^{q_{i}} (Ω)$ and $∥ D^{γ_{i}} w_{i} ∥_{L^{q_{i}}} \leq C ∥ w_{i} ∥_{W^{k, p} (Ω)}$ as follows: $\frac{1}{q_{i}} = 0$ when $s_{i} p > n$ ; $\frac{1}{q_{i}} = \frac{1}{p} - \frac{s_{i}}{n}$ when $s_{i} p < n$ ; and $\frac{1}{q_{i}} = \frac{1}{2 n}$ when $s_{i} p = n$ , which is available because the critical embedding supplies every finite exponent. In every case $\frac{1}{q_{1}} + \frac{1}{q_{2}} \leq \frac{1}{p}$ : for two subcritical exponents this is $\frac{2}{p} - \frac{s_{1} + s_{2}}{n} \leq \frac{2}{p} - \frac{k}{n} < \frac{1}{p}$ because $k p > n$ ; for one critical and one subcritical exponent it is $\frac{1}{2 n} + \frac{1}{p} - \frac{s}{n} \leq \frac{1}{p}$ because $s \geq 1$ ; for two critical exponents it is $\frac{1}{n} \leq \frac{1}{p}$ because $p \leq n$ (recall $s_{i} p = n$ with $s_{i} \geq 1$ forces $p \leq n$ ); and a supercritical exponent contributes $0$ . Hence by generalized Holder [F3], $D^{γ_{1}} w_{1} D^{γ_{2}} w_{2} \in L^{p} (Ω)$ with $∥ D^{γ_{1}} w_{1} D^{γ_{2}} w_{2} ∥_{L^{p}} \leq C (n, p, k, Ω) ∥ w_{1} ∥_{W^{k, p}} ∥ w_{2} ∥_{W^{k, p}}$ .

2.1F2F4F5step 1.1givenalgebra∎

The Leibniz identity and the algebra bound. By [F2] choose $u_{j}, v_{j} \in C^{\infty} (Ω) \cap W^{k, p} (Ω)$ with $u_{j} \to u$ and $v_{j} \to v$ in $W^{k, p} (Ω)$ . Fix $∣ α ∣ \leq k$ ; for the smooth factors, repeated classical differentiation gives the finite Leibniz formula, and [F4] identifies its classical derivatives with weak derivatives: $D^{α} (u_{j} v_{j}) = \sum_{β \leq α} (\binom{α}{β}) D^{β} u_{j} D^{α - β} v_{j}$ . Applying step 1.1 to the pairs $(u_{j} - u, v_{j})$ and $(u, v_{j} - v)$ with $(γ_{1}, γ_{2}) = (β, α - β)$ shows that each summand converges in $L^{p} (Ω)$ to $D^{β} u D^{α - β} v$ , and the case $α = 0$ gives $u_{j} v_{j} \to u v$ in $L^{p} (Ω)$ ; therefore, for every test function $φ \in C_{c}^{\infty} (Ω)$ , $\int_{Ω} u v \partial^{α} φ = \lim_{j} \int_{Ω} u_{j} v_{j} \partial^{α} φ = (- 1)^{∣ α ∣} \lim_{j} \int_{Ω} D^{α} (u_{j} v_{j}) φ = (- 1)^{∣ α ∣} \int_{Ω} g_{α} φ$ with $g_{α} : = \sum_{β \leq α} (\binom{α}{β}) D^{β} u D^{α - β} v \in L^{p} (Ω)$ . By the characterization [F4], $g_{α}$ is the weak $α$ -derivative of $u v$ for every $∣ α ∣ \leq k$ , so $u v \in W^{k, p} (Ω)$ , and $∥ u v ∥_{W^{k, p} (Ω)} \leq \sum_{∣ α ∣ \leq k} ∥ g_{α} ∥_{L^{p}} \leq C (n, p, k, Ω) ∥ u ∥_{W^{k, p}} ∥ v ∥_{W^{k, p}}$ by step 1.1, which is the asserted algebra inequality. Completeness [F5] makes it a Banach algebra with continuous multiplication (after an equivalent norm rescaling if a submultiplicative norm is required).

Source notes

The algebra property of $W^{k, p}$ above the critical index is the standard consequence of the higher-order embedding and the Leibniz rule; Kinnunen's Morrey theorem and higher-order iteration, together with Hunter's first-order embedding, supply the context; the product estimate and weak Leibniz passage are reconstructed here. The proof above isolates the two ingredients: the product estimate 1.1, where the embedding either makes a factor bounded (when its remaining order exceeds $n / p$ ), supplies every finite exponent (at the critical order) or supplies the Sobolev exponent (below it), and Holder combines the two; and the Leibniz identity 2.1, which passes the classical formula for smooth approximations to the limit in $L^{p}$ and identifies the limit through the test-function definition of the weak derivative.

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The $p = n$ endpoint: no $L^{\infty}$ or Holder embedding

Remark

Let $n \geq 2$ . On nonempty bounded $C^{1}$ domains in $R^{n}$ (or domains satisfying the all-exponents extension hypothesis of the critical embedding theorem), $W^{1, n}$ embeds into $L^{q}$ for every finite $q$ but not into $L^{\infty}$ , and some $W^{1, n}$ classes have no representative in $C^{0, α} (\overline{Ω})$ for any $α > 0$ . The exponential improvement of Trudinger-Moser and the BMO/John-Nirenberg route to the same integrability are outside this pair's scope and are recorded in the planning scope-denial ledger; the companion page carries the witnesses.

The dimension restriction is essential: on a bounded interval $I$ , every $W^{1, 1} (I)$ class has an absolutely continuous representative and satisfies $∥ u ∥_{L^{\infty} (I)} \leq ∣ I ∣^{- 1} ∥ u ∥_{L^{1} (I)} + ∥ u^{'} ∥_{L^{1} (I)}$ .

Source notes

The endpoint limitation is Kinnunen's Remark 3.16 and the surrounding discussion of the critical exponent, with Hunter's Example 3.30 as the explicit logarithmic witness with Laugesen's Theorem 3.23 supplying the supercritical comparison. This remark records the limitation only: the positive embedding $W^{1, n} ↪ L^{q}$ for finite $q$ and the failure of the $L^{\infty}$ and Holder bounds are proved elsewhere on this page and on its companion, and the Trudinger-Moser and BMO routes are deliberately not built in this pair.

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Domain classes covered by the mean-zero Poincare inequality

Remark

This page proves the mean-zero Poincare-Wirtinger inequality directly for bounded John domains The mean-zero Poincare inequality on bounded John domains and for bounded convex domains Poincare-Wirtinger on bounded convex domains by the direct pairwise argument, both on the same page. Bounded $C^{1}$ and bounded Lipschitz domains satisfy the uniform interior cone condition, and every nonempty bounded connected open set satisfying that uniform condition is a John domain (Kinnunen, Remark 5.32(1)); hence the inequality applies to those classes as well. The John-class hypothesis $1 \leq p < \infty$ is the one used by the direct proof; no Sobolev exponent and no compactness input enter it. The bounded connected extension-domain Sobolev-Poincare form $1 < p < n$ is a separate statement on this page, proved from the local mean-zero estimate and the extension-domain embedding, and it does not supersede the John-domain result.

Source notes

The class inclusions and the "twisted cones" picture are Kinnunen's Remark 5.32, printed p. 141, with Laugesen's Theorem 3.29 supplying the bounded connected smooth-domain Poincare comparison. This remark records scope only: it asserts no inequality of its own, and the two direct John and convex results together with the separate extension-domain statement are proved elsewhere on this page.

5 · Examples, counterexamples and false statements

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Sobolev Poincare and Morrey Inequalities

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

The Sobolev conjugate exponent and the scaling identity

Definition

Pointwise potential bound for compactly supported smooth functions

Statement

Facts & Assumptions

Proof

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The p=1 Gagliardo-Nirenberg-Sobolev inequality

Statement

Facts & Assumptions

Proof

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The Gagliardo-Nirenberg-Sobolev inequality for 1<p<n1<p<n

Statement

Facts & Assumptions

Proof

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The Sobolev inequality for zero-boundary Sobolev closures on open sets

Statement

Facts & Assumptions

Proof

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Poincare inequality on a ball

Statement

Facts & Assumptions

Proof

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The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction

Statement

Facts & Assumptions

Proof

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Mean-zero Poincare estimate on bounded connected extension domains below the dimension

Statement

Facts & Assumptions

Proof

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John domains and the John constant

Definition

Bounded-overlap ball chains in a bounded John domain

Statement

Facts & Assumptions

Proof

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The truncated Riesz kernel is bounded on LpLp of a bounded set

Statement

Facts & Assumptions

Proof

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The mean-zero Poincare inequality on bounded John domains

Statement

Facts & Assumptions

Proof

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The Poincare inequality with a positive-measure zero set

Statement

Facts & Assumptions

Proof

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Poincare-Wirtinger on bounded convex domains by the direct pairwise argument

Statement

Facts & Assumptions

Proof

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Sobolev embedding on bounded extension domains for p<np<n

Statement

Facts & Assumptions

Proof

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Sobolev-Poincare on bounded connected extension domains

Statement

Facts & Assumptions

Proof

Source notes

The critical Sobolev embedding into every finite LqLq

The Gagliardo-Nirenberg-Sobolev inequality for $1 < p < n$

The truncated Riesz kernel is bounded on $L^{p}$ of a bounded set

Sobolev embedding on bounded extension domains for $p < n$

The critical Sobolev embedding into every finite $L^{q}$

Morrey's inequality for $p > n$

$W^{1, \infty}$ functions on convex domains have Lipschitz representatives

The Sobolev space $W^{k, p}$ is an algebra above the critical index

The $p = n$ endpoint: no $L^{\infty}$ or Holder embedding