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Fractional Hilbert Sobolev algebra on a smooth exterior domain

Statement

Assume the Axiom of Choice. Let $n \in {2, 3}$ , $n / 2 < s \leq 2$ , and let $Ω$ be the complement of a compact obstacle with smooth boundary. Write $H^{s} (Ω)$ for restrictions to $Ω$ of whole-space Bessel-potential functions, with the infimum of the whole-space $H^{s}$ norms over extensions as its norm. Then there is a constant $C$ depending on $n, s, Ω$ such that $∥ u ∥_{L^{\infty} (Ω)} \leq C ∥ u ∥_{H^{s} (Ω)}$ and $∥ u v ∥_{H^{s} (Ω)} \leq C ∥ u ∥_{H^{s} (Ω)} ∥ v ∥_{H^{s} (Ω)}$ .

For a complex polynomial $P (z, \overset{ˉ}{z})$ of degree at most five with $P (0, 0) = 0$ , there is a constant $C_{P}$ such that $∥ P (u, \overset{ˉ}{u}) ∥_{H^{s}} \leq C_{P} \sum_{j = 1}^{5} ∥ u ∥_{H^{s}}^{j}$ and $∥ P (u, \overset{ˉ}{u}) - P (v, \overset{ˉ}{v}) ∥_{H^{s}} \leq C_{P} ∥ u - v ∥_{H^{s}} \sum_{j = 1}^{5} (∥ u ∥_{H^{s}} + ∥ v ∥_{H^{s}})^{j - 1}$ . In particular this covers cubic and quintic nonlinearities. The Dirichlet trace means the $H^{1}$ trace on the obstacle boundary, defined using a bounded smooth collar. If both factors have zero Dirichlet trace, their product has zero Dirichlet trace. No endpoint assertion at $s = n / 2$ is made.

Facts & Assumptions

Given: The hypotheses, spaces and conventions in the Statement.

[F1]

The Axiom of Choice supplies the countable-choice assumptions of the Fourier and convolution inputs. The Axiom of Choice

[F2]

Whole-space $H^{s}$ is the completion of Schwartz functions with norm $∥ ⟨ ξ ⟩^{s} \hat{f} ∥_{2}$ , where $⟨ ξ ⟩ = (1 + ∣ ξ ∣^{2})^{1 / 2}$ and the Fourier transform uses the negative-sign $2 π$ convention. Schwartz functions are dense in this norm for the present $s$ . Real-order Hˢ as weighted Fourier distributions Schwartz functions are dense in the Fourier Bessel-potential norm

[F3]

For Schwartz functions Fourier inversion is absolutely convergent, and the transform of their product is the convolution of their transforms. Holder's inequality and Young's $L^{1} * L^{2} \to L^{2}$ inequality apply to the indicated nonnegative functions. Fourier inversion on Schwartz space Fourier transform of a product with one integrable transform Holder's inequality for integrals, including the endpoint cases Young's convolution inequality under Countable Choice

[F4]

On the smooth exterior, restriction-quotient spaces have the compatible collar and free-tail extension/retraction described by Simultaneous Sobolev extension and localization. For a bounded smooth collar $G \subset Ω$ whose boundary contains the obstacle boundary $B = \partial Ω$ , the $p = 2$ , $K = C$ instance of The Lᵖ trace operator on a bounded C¹ domain gives a bounded $T_{G} : W^{1, 2} (G; C) \to L^{2} (\partial G; C)$ agreeing with classical restriction on continuous Sobolev functions. Restrict its output to $B$ and write $T u = (T_{G} (u ∣_{G})) ∣_{B}$ ; restriction to these boundary components does not increase the $L^{2}$ norm. This is the same trace operator appearing in The sharp trace theorem: boundedness and range in the fractional space.

[F5]

Fourier transformation is unitary on $L^{2}$ , and distributional differentiation corresponds to multiplication by $2 π i ξ_{j}$ . Plancherel theorem Distributional derivatives are polynomial Fourier multipliers

[F6]

Schwartz functions are actual smooth complex functions $f$ for which $\sup_{x \in R^{n}} ∣ x^{α} \partial^{β} f (x) ∣ < \infty$ for every pair of multi-indices $α, β$ . Schwartz space and its seminorms

[F7]

For any open set $D$ , the weak $H^{1} (D)$ notation means $W^{1, 2} (D; C)$ , consisting of $L^{2}$ classes with first weak derivatives in $L^{2}$ , with norm $∥ h ∥_{W^{1, 2} (D)} = (∥ h ∥_{2}^{2} + \sum_{k = 1}^{n} ∥ \partial_{k} h ∥_{2}^{2})^{1 / 2}$ . Under the assumed AC this space is complete. Integer-order Sobolev spaces and their norms Integer-order Sobolev spaces are Banach

Proof

Given: The hypotheses, spaces and conventions in the Statement.

[F1] The Axiom of Choice supplies the countable-choice assumptions of the Fourier and convolution inputs. The Axiom of Choice

[F2] Whole-space $H^{s}$ is the completion of Schwartz functions with norm $∥ ⟨ ξ ⟩^{s} \hat{f} ∥_{2}$ , where $⟨ ξ ⟩ = (1 + ∣ ξ ∣^{2})^{1 / 2}$ and the Fourier transform uses the negative-sign $2 π$ convention. Schwartz functions are dense in this norm for the present $s$ . Real-order Hˢ as weighted Fourier distributions Schwartz functions are dense in the Fourier Bessel-potential norm

[F3] For Schwartz functions Fourier inversion is absolutely convergent, and the transform of their product is the convolution of their transforms. Holder's inequality and Young's $L^{1} * L^{2} \to L^{2}$ inequality apply to the indicated nonnegative functions. Fourier inversion on Schwartz space Fourier transform of a product with one integrable transform Holder's inequality for integrals, including the endpoint cases Young's convolution inequality under Countable Choice

[F4] On the smooth exterior, restriction-quotient spaces have the compatible collar and free-tail extension/retraction described by Simultaneous Sobolev extension and localization. For a bounded smooth collar $G \subset Ω$ whose boundary contains the obstacle boundary $B = \partial Ω$ , the $p = 2$ , $K = C$ instance of The Lᵖ trace operator on a bounded C¹ domain gives a bounded $T_{G} : W^{1, 2} (G; C) \to L^{2} (\partial G; C)$ agreeing with classical restriction on continuous Sobolev functions. Restrict its output to $B$ and write $T u = (T_{G} (u ∣_{G})) ∣_{B}$ ; restriction to these boundary components does not increase the $L^{2}$ norm. This is the same trace operator appearing in The sharp trace theorem: boundedness and range in the fractional space.

[F5] Fourier transformation is unitary on $L^{2}$ , and distributional differentiation corresponds to multiplication by $2 π i ξ_{j}$ . Plancherel theorem Distributional derivatives are polynomial Fourier multipliers

[F6] Schwartz functions are actual smooth complex functions $f$ for which $\sup_{x \in R^{n}} ∣ x^{α} \partial^{β} f (x) ∣ < \infty$ for every pair of multi-indices $α, β$ . Schwartz space and its seminorms

[F7] For any open set $D$ , the weak $H^{1} (D)$ notation means $W^{1, 2} (D; C)$ , consisting of $L^{2}$ classes with first weak derivatives in $L^{2}$ , with norm $∥ h ∥_{W^{1, 2} (D)} = (∥ h ∥_{2}^{2} + \sum_{k = 1}^{n} ∥ \partial_{k} h ∥_{2}^{2})^{1 / 2}$ . Under the assumed AC this space is complete. Integer-order Sobolev spaces and their norms Integer-order Sobolev spaces are Banach

Given: The hypotheses and conventions in the Statement.

Goal: Prove the embedding, multiplication and polynomial estimates, and the trace assertion.

1.1F2F3F6given

Since $2 s > n$ , polar integration gives $A = ∥ ⟨ ξ ⟩^{- s} ∥_{2} < \infty$ . For Schwartz $f$ , Holder gives $∥ \hat{f} ∥_{1} \leq A ∥ f ∥_{H^{s}}$ , and inversion gives $∥ f ∥_{\infty} \leq A ∥ f ∥_{H^{s}}$ . Moreover $⟨ ξ ⟩^{s} \leq C_{s} (⟨ η ⟩^{s} + ⟨ ξ - η ⟩^{s})$ : use $⟨ ξ ⟩ \leq ⟨ η ⟩ + ⟨ ξ - η ⟩$ and the elementary inequality for positive powers. The product formula and Young's inequality therefore give $∥ f g ∥_{H^{s}} \leq C_{s} (∥ ⟨ \cdot ⟩^{s} \hat{f} ∥_{2} ∥ \hat{g} ∥_{1} + ∥ \hat{f} ∥_{1} ∥ ⟨ \cdot ⟩^{s} \hat{g} ∥_{2}) \leq 2 C_{s} A ∥ f ∥_{H^{s}} ∥ g ∥_{H^{s}}$ .

2.1F2F3F6step 1.1

For arbitrary whole-space $H^{s}$ elements choose Schwartz approximating sequences. The preceding estimates make them Cauchy both uniformly and in the product norm. The uniform limits represent the original $L^{2}$ classes, since $∥ h ∥_{2} \leq ∥ h ∥_{H^{s}}$ ; consequently the limiting product is their pointwise product almost everywhere. Bilinearity and the difference identity $f_{j} g_{j} - f_{k} g_{k} = (f_{j} - f_{k}) g_{j} + f_{k} (g_{j} - g_{k})$ give the same estimates for all whole-space factors. Complex conjugation preserves the norm because its Fourier transform is the conjugate reflected transform.

3.1F2F4step 2.1given

For $u, v \in H^{s} (Ω)$ choose whole-space extensions $U, V$ . Restriction of $U V$ is $u v$ , so the quotient norm and step 2.1 give $∥ u v ∥_{H^{s} (Ω)} \leq C ∥ U ∥_{H^{s} (R^{n})} ∥ V ∥_{H^{s} (R^{n})}$ . Taking infima independently over the two nonempty extension classes gives the product estimate. The same infimum argument gives the $L^{\infty}$ estimate, and conjugating extensions shows $∥ \overset{ˉ}{u} ∥_{H^{s} (Ω)} = ∥ u ∥_{H^{s} (Ω)}$ .

3.2F2F5F6F7step 1.1step 2.1given

Let $U_{j}, V_{j}$ be Schwartz approximations to extensions $U, V$ of $u, v$ . In the weak Sobolev convention of [F7], the Fourier derivative and Plancherel identities give $∥ h ∥_{W^{1, 2} (R^{n})}^{2} = ∥ \hat{h} ∥_{2}^{2} + 4 π^{2} \sum_{k = 1}^{n} ∥ ξ_{k} \hat{h} ∥_{2}^{2} \leq C ∥ h ∥_{H^{s} (R^{n})}^{2}$ for Schwartz $h$ , because $s > 1$ . This bound and step 2.1 make $U_{j}, V_{j}, U_{j} V_{j}$ Cauchy in $W^{1, 2} (R^{n}; C)$ . Completeness from [F7] gives limits in that space, and their $L^{2}$ limits identify them with $U, V, U V$ , respectively. Restriction decreases the integrals defining the weak Sobolev norm, so the restrictions and products converge in weak $H^{1}$ on a bounded smooth collar containing the obstacle boundary. They also converge uniformly by steps 1.1 and 2.1.

4.1step 3.1algebra

Apply the product estimate repeatedly to each monomial $u^{a} {\overset{ˉ}{u}}^{b}$ , where $1 \leq a + b \leq 5$ . For differences telescope the $a + b$ factors, replacing one factor at a time; each difference is either $u - v$ or its conjugate. Summing the finitely many monomials gives both displayed polynomial inequalities, with a constant depending on the coefficients and the algebra constant. The zero constant term is needed because the exterior has infinite measure.

4.2F4step 3.2

Apply the bounded-domain $L^{2}$ trace operator $T_{G}$ from F4 to the collar $G$ and restrict its values to the obstacle boundary $B$ . Step 3.2 and its boundedness give $T (U_{j} ∣_{Ω}) \to T u$ , $T (V_{j} ∣_{Ω}) \to T v$ and $T ((U_{j} V_{j}) ∣_{Ω}) \to T (u v)$ in $L^{2} (B)$ . Here $T (U_{j} ∣_{Ω}) = (T_{G} (U_{j} ∣_{G})) ∣_{B}$ , with the analogous notation for the other factors and products. Since the smooth restrictions belong to $C (\overline{G}) \cap W^{1, 2} (G; C)$ , the classical-restriction property in F4 gives $T (U_{j} V_{j}) = T (U_{j}) T (V_{j})$ on $B$ . Uniform convergence on the compact boundary and convergence of the other trace in $L^{2}$ show that these products converge to $(T u) (T v)$ in $L^{2}$ . Hence $T (u v) = (T u) (T v)$ ; in particular it vanishes if both traces vanish. The bounded-collar definition is independent of the chosen collar because smooth restrictions agree on the common obstacle boundary and are dense in the relevant approximations.

5.1step 1.1step 3.1step 4.1step 4.2∎

Steps 3.1 and 4.1 establish the embedding, algebra and polynomial estimates, and step 4.2 establishes the zero-trace consequence. The integrability used in step 1.1 requires $2 s > n$ , so the proof makes no endpoint claim.

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Remark 5: controllability above the Sobolev algebra threshold

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