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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 $1 / σ (S^{n - 1})$ .

Depends on

Used by

W^1,∞ functions on convex domains have Lipschitz representatives Theorem

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Sources

Juha Kinnunen, Sobolev Spaces (Aalto University, 2026, complete graduate lecture notes) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter notes) (standard reference, not scraped)