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Independent exterior fractional Sobolev embedding

Statement

Assume the Axiom of Choice. Let $Ω \subset R^{n}$ be a smooth exterior domain, meaning the complement of a compact set with smooth boundary, and let $0 < a < 1$ , $1 < q < \infty$ and $a q > n$ . With the complex-interpolation convention $W^{a, q} (Ω) = [L^{q} (Ω), W^{1, q} (Ω)]_{a}$ of An explicit common interpolation model for the fractional Strichartz targets, there is a constant $C$ such that $∥ u ∥_{L^{\infty} (Ω)} \leq C ∥ u ∥_{W^{a, q} (Ω)}$ for every $u \in W^{a, q} (Ω)$ .

In particular, the hypotheses hold when $n = 2$ , $p > 2$ , $a = 1 - 1 / p$ and $q = 2 p / (p - 2)$ .

External input

For $0 < a < 1$ , $1 < q < \infty$ and $a q > n$ , the cited whole-space theorem continuously embeds $H^{a, q} (R^{n})$ into bounded Hölder-continuous functions. Here $H^{a, q}$ denotes the whole-space Bessel-potential space, equivalently $[L^{q} (R^{n}), W^{1, q} (R^{n})]_{a}$ with equivalent norms. This input makes no exterior-domain assertion.

Facts & Assumptions

Given: The hypotheses and notation of the Statement.

[F1]

The Axiom of Choice is assumed as in The Axiom of Choice and in the interpolation-model supplier.

[F2]

By An explicit common interpolation model for the fractional Strichartz targets, under these hypotheses there is a bounded extension $E : W^{a, q} (Ω) \to H^{a, q} (R^{n})$ such that $(E u) ∣_{Ω} = u$ almost everywhere. Its whole-space Bessel and complex-interpolation norms are equivalent.

[F3]

The independent whole-space input in the External input section provides a constant $C_{0}$ such that $∥ F ∥_{L^{\infty} (R^{n})} \leq C_{0} ∥ F ∥_{H^{a, q} (R^{n})}$ for every $F \in H^{a, q} (R^{n})$ .

[F4]

For $D = Ω$ or $D = R^{n}$ , the norm $∥ v ∥_{L^{\infty} (D)}$ is the essential supremum of the real-valued measurable function $∣ v ∣$ with respect to Lebesgue measure on $D$ , using The essential supremum of a measurable function with respect to a measure. Thus it is the infimum of the bounds $M \in [0, \infty]$ satisfying $∣ v ∣ \leq M$ almost everywhere on $D$ .

Proof

Given: The hypotheses and notation of the Statement.

[F1] The Axiom of Choice is assumed as in The Axiom of Choice and in the interpolation-model supplier.

[F2] By An explicit common interpolation model for the fractional Strichartz targets, under these hypotheses there is a bounded extension $E : W^{a, q} (Ω) \to H^{a, q} (R^{n})$ such that $(E u) ∣_{Ω} = u$ almost everywhere. Its whole-space Bessel and complex-interpolation norms are equivalent.

[F3] The independent whole-space input in the External input section provides a constant $C_{0}$ such that $∥ F ∥_{L^{\infty} (R^{n})} \leq C_{0} ∥ F ∥_{H^{a, q} (R^{n})}$ for every $F \in H^{a, q} (R^{n})$ .

Given: The hypotheses and data in the Statement.

Goal: Establish the asserted bounded inclusion and verify the stated parameter example.

1.1F1F2given

For $u \in W^{a, q} (Ω)$ , set $F = E u$ . The extension in [F2] gives $F ∣_{Ω} = u$ almost everywhere and $∥ F ∥_{H^{a, q} (R^{n})} \leq C_{E} ∥ u ∥_{W^{a, q} (Ω)}$ for a constant $C_{E}$ independent of $u$ .

2.1F3F4step 1.1

If $M$ is a finite almost-everywhere bound for $∣ F ∣$ on $R^{n}$ , its exceptional null set restricts to a Lebesgue-null set in $Ω$ . Since $F ∣_{Ω} = u$ almost everywhere by step 1.1, adjoining the null set where this equality fails shows $∣ u ∣ \leq M$ almost everywhere on $Ω$ . Thus every such bound for $F$ is a bound for $u$ . Taking infima in the definition in [F4] gives $∥ u ∥_{L^{\infty} (Ω)} \leq ∥ F ∥_{L^{\infty} (R^{n})}$ ; finite bounds exist by [F3]. Consequently the whole-space bound in [F3] and step 1.1 give $∥ u ∥_{L^{\infty} (Ω)} \leq ∥ F ∥_{L^{\infty} (R^{n})} \leq C_{0} C_{E} ∥ u ∥_{W^{a, q} (Ω)}$ . This proves the continuous inclusion.

3.1step 2.1algebragiven∎

If $p > 2$ , then $0 < 1 - 1 / p < 1$ and $1 < 2 p / (p - 2) < \infty$ . Moreover $a q = 2 (p - 1) / (p - 2) = 2 + 2 / (p - 2) > 2 = n$ , so the same inclusion applies to the stated parameters.

Source notes

The whole-space embedding is an imported premise; the exterior restriction argument above is local. The published integer-order transfer result does not itself cover this fractional interpolation model, and the published fractional inequality inspected concerns the subcritical Slobodeckij setting.

Depends on

Used by · 1

Two-dimensional exponents removing the Strichartz loss

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

José C. Bellido and Guillermo García-Sáez, Bessel Potential Spaces and Complex Interpolation: Continuous embeddings, arXiv:2503.04310v6 · Definition 3.6 and Theorem 3.7, PDF p. 17; Theorem 3.21(3), PDF pp. 25–26

For 0<a<1, 1<q<infinity and aq>n, the whole-space complex-interpolation space H^{a,q}(R^n) embeds continuously into C^{0,mu}(R^n) for 0<mu<=a-n/q, and hence into L-infinity. The interpolation and Bessel-potential definitions have equivalent norms.

Hypotheses: R^n, n>=1, 0<a<1, 1<q<infinity and aq>n. The cited embedding is a whole-space result; it does not itself assert an exterior-domain extension theorem.

Reading limits: Read the v6 PDF statements of Definition 3.6, Theorems 3.7 and 3.21(3). The embedding proof and its underlying interpolation results were not reconstructed or independently checked.

Sources

José C. Bellido and Guillermo García-Sáez, Bessel Potential Spaces and Complex Interpolation: Continuous embeddings, arXiv:2503.04310v6 · Definition 3.6 and Theorem 3.7, PDF p. 17; Theorem 3.21(3), PDF pp. 25–26