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The critical fractional Sobolev inequality on $R^{d}$

Statement

Assume the Axiom of Countable Choice. Let $d \geq 1$ , $0 < θ < 1$ , $1 \leq p < \infty$ with $p θ < d$ and $p_{⋆} : = \frac{d p}{d - p θ}$ . There is $C = C (d, p, θ) > 0$ such that every measurable, compactly supported $f : R^{d} \to R$ satisfies $∥ f ∥_{L^{p_{⋆}} (R^{d})}^{p} \leq C [f]_{θ, p}^{p} .$ Consequently, for every bounded open $U \subseteq R^{d}$ and every $1 \leq q \leq p_{⋆}$ there is $C^{'} = C^{'} (d, p, θ, U, q)$ with $∥ f ∥_{L^{q} (U)} \leq C^{'} (∥ f ∥_{L^{p} (R^{d})} + [f]_{θ, p})$ for every compactly supported measurable $f$ .

Facts & Assumptions

Given: Countable Choice, $d \geq 1$ , $0 < θ < 1$ , $1 \leq p < \infty$ with $p θ < d$ , and $p_{⋆} = d p / (d - p θ)$ ; write $α : = p θ / d$ , so that $p / p_{⋆} = (d - p θ) / d = 1 - α$ and $p_{⋆} / p = 1 / (1 - α)$ .

[F1]

Dyadic summability. For a bounded nonnegative nonincreasing sequence $(a_{k})$ vanishing for all large $k$ and $T = 2^{p} > 1$ , $\sum_{k} a_{k}^{1 - α} T^{k} \leq C_{1} \sum_{k : a_{k} \neq 0} a_{k + 1} a_{k}^{- α} T^{k}$ . (A dyadic summability estimate for decreasing level-set sequences)

[F2]

Level-set bound. For $f \in L^{\infty}$ compactly supported with $a_{k} = ∣ {∣ f ∣ > 2^{k}} ∣$ , $[f]_{θ, p}^{p} \geq c \sum_{k : a_{k} \neq 0} a_{k + 1} a_{k}^{- α} 2^{p k}$ . (The Slobodeckij seminorm bounds the dyadic level-set sum)

[F3]

Fatou and dominated convergence. Fatou's lemma bounds the integral of a pointwise limit below by the lower limit of the integrals; dominated convergence applies under an integrable dominating function. (Fatou's lemma, Dominated convergence)

[F4]

Interpolation and H"older. For $p < q < p_{⋆}$ , let $σ \in (0, 1)$ be defined by $1 / q = (1 - σ) / p + σ / p_{⋆}$ . Lyapunov interpolation, with its parameter $1 - σ$ , gives $∥ g ∥_{L^{q}} \leq ∥ g ∥_{L^{p}}^{1 - σ} ∥ g ∥_{L^{p_{⋆}}}^{σ}$ . For $q \leq p$ on a set $U$ of finite measure, $∥ g ∥_{L^{q} (U)} \leq ∣ U ∣^{1 / q - 1 / p} ∥ g ∥_{L^{p} (U)}$ . (Lyapunov interpolation inequality for $L^{p}$ norms, Holder's inequality for integrals, including the endpoint cases)

[F5]

Seminorm and classes. $[\cdot]_{θ, p}$ is the Slobodeckij seminorm, finite on $W^{θ, p}$ . The scalar truncation $T_{N} (t) = \max (- N, \min (N, t))$ is 1-Lipschitz, hence $∣ T_{N} (f (x)) - T_{N} (f (y)) ∣ \leq ∣ f (x) - f (y) ∣$ and $[T_{N} f]_{θ, p} \leq [f]_{θ, p}$ . (The Gagliardo--Slobodeckij space on Euclidean space, The space $L^{p} (μ)$ as the quotient by null functions)

Proof

technique · prove the inequality for bounded compactly supported $f$ by the layer-cake expansion and the two dyadic estimates, then pass to general $f$ by truncation and Fatou. If the seminorm is infinite, the inequality is immediate

1.1F1F2F5algebra

Let first $f \in L^{\infty}$ have compact support, put $A_{k} = {∣ f ∣ > 2^{k}}$ and $a_{k} = ∣ A_{k} ∣$ . On $D_{k} = A_{k} ∖ A_{k + 1}$ one has $∣ f ∣ \leq 2^{k + 1}$ , and the $D_{k}$ partition ${f \neq 0}$ , and $f$ vanishes on the remaining set, so $∥ f ∥_{p_{⋆}}^{p_{⋆}} = \int ∣ f ∣^{p_{⋆}} \leq \sum_{k} 2^{(k + 1) p_{⋆}} ∣ D_{k} ∣ \leq 2^{p_{⋆}} \sum_{k} 2^{k p_{⋆}} a_{k}$ ; raising to the power $p / p_{⋆} < 1$ and using the concavity bound $(\sum_{k} b_{k}^{1 / (1 - α)})^{1 - α} \leq \sum_{k} b_{k}$ for $b_{k} : = a_{k}^{1 - α} 2^{p k}$ , whose $1 / (1 - α)$ -th powers are $a_{k} 2^{k p_{⋆}}$ , gives $∥ f ∥_{p_{⋆}}^{p} \leq 2^{p} \sum_{k} a_{k}^{1 - α} 2^{p k}$ . Since the sequence $a_{k}$ is bounded, nonincreasing and eventually $0$ , [F1] followed by [F2] bounds the last sum by a constant times $[f]_{θ, p}^{p}$ , proving the inequality for this $f$ .

2.1F3F5step 1.1

For general compactly supported measurable $f$ with $[f]_{θ, p} < \infty$ put $f_{N} : = \max {- N, \min {N, f}}$ . Then $f_{N} \to f$ pointwise with $∣ f_{N} ∣ \leq ∣ f ∣$ , so $[f_{N}]_{θ, p} \leq [f]_{θ, p}$ by the pointwise contraction in [F5]; the bounded case of step 1.1 gives $∥ f_{N} ∥_{p_{⋆}}^{p} \leq C [f_{N}]_{θ, p}^{p} \leq C [f]_{θ, p}^{p}$ , and Fatou's lemma [F3] passes to the limit: $∥ f ∥_{p_{⋆}}^{p} \leq {lim inf}_{N} ∥ f_{N} ∥_{p_{⋆}}^{p} \leq C [f]_{θ, p}^{p}$ , which is the asserted inequality.

3.1F1F2F3F4step 1.1step 2.1algebra∎

Let $U$ be bounded open and $1 \leq q \leq p_{⋆}$ . If $U = \emptyset$ or $∥ f ∥_{L^{p} (R^{d})} + [f]_{θ, p} = \infty$ , the conclusion is immediate. For $q \leq p$ , Holder [F4] on the finite-measure set $U$ gives $∥ f ∥_{L^{q} (U)} \leq ∣ U ∣^{1 / q - 1 / p} ∥ f ∥_{L^{p} (U)} \leq ∣ U ∣^{1 / q - 1 / p} ∥ f ∥_{L^{p} (R^{d})}$ . If $p < q < p_{⋆}$ , choose $σ \in (0, 1)$ so that $1 / q = (1 - σ) / p + σ / p_{⋆}$ . Lyapunov [F4], the embedding of the restricted $L^{p}$ and $L^{p_{⋆}}$ norms below their global norms, and step 2.1 give $∥ f ∥_{L^{q} (U)} \leq ∥ f ∥_{L^{p} (R^{d})}^{1 - σ} (C^{1 / p} [f]_{θ, p})^{σ} .$ Put $a = ∥ f ∥_{L^{p} (R^{d})}$ and $b = C^{1 / p} [f]_{θ, p}$ . Weighted AM--GM yields $a^{1 - σ} b^{σ} \leq (1 - σ) a + σ b \leq \max {1, C^{1 / p}} (a + [f]_{θ, p})$ . At $q = p_{⋆}$ , step 2.1 directly gives $∥ f ∥_{L^{p_{⋆}} (U)} \leq C^{1 / p} [f]_{θ, p}$ . These estimates, and the $q \leq p$ Holder bound, give the claimed consequence with a constant depending on $d, p, θ, U, q$ . Countable Choice is inherited through [F1], [F2] and [F3].

Depends on

Used by

Subcritical compactness for compactly supported Slobodeckij functions Theorem

Dependency tree · two levels

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Sources

Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces (arXiv:1104.4345, survey) (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)

The critical fractional Sobolev inequality on RdRd