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Bessel potentials shift Sobolev order

Statement

Assume Countable Choice. Let $n \geq 1$ , $s, t \in R$ , let $E_{σ} : H^{σ} (R^{n}) \to S^{'} (R^{n})$ be the canonical embedding of the real-order Bessel-potential completion, and let $⟨ D ⟩^{t}$ and $(I - Δ)^{t / 2}$ be the distributional Fourier multipliers of Japanese-bracket and Laplacian Bessel-potential operators. For $U \in H^{s}$ let $g = ⟨ ξ ⟩^{s} F (E_{s} U) \in L^{2} (R^{n})$ be its weighted Fourier class, so that $∥ U ∥_{H^{s}} = ∥ g ∥_{2}$ (Real-order H^s as weighted Fourier distributions). Then:

The bracket operator shifts order isometrically. $Φ_{t} (U) : = E_{s - t}^{- 1} (⟨ D ⟩^{t} E_{s} U), U \in H^{s},$ defines a surjective complex-linear isometry $Φ_{t} : H^{s} (R^{n}) \to H^{s - t} (R^{n})$ with $∥ Φ_{t} (U) ∥_{H^{s - t}} = ∥ U ∥_{H^{s}}$ for every $U$ , whose weighted Fourier class in $H^{s - t}$ is again $g$ ; its inverse is $Φ_{- t}$ .
The Laplacian operator shifts order with equivalent norms. With $r_{t} (ξ) = (\frac{1 + 4 π^{2} ∣ ξ ∣^{2}}{1 + ∣ ξ ∣^{2}})^{t / 2}, c_{t} = \min (1, (2 π)^{t}), C_{t} = \max (1, (2 π)^{t}),$ the map $Ψ_{t} (U) : = E_{s - t}^{- 1} ((I - Δ)^{t / 2} E_{s} U)$ defines a bounded complex-linear bijection $H^{s} (R^{n}) \to H^{s - t} (R^{n})$ whose weighted Fourier class is $r_{t} g$ , with $c_{t} ∥ U ∥_{H^{s}} \leq ∥ Ψ_{t} (U) ∥_{H^{s - t}} \leq C_{t} ∥ U ∥_{H^{s}} (U \in H^{s}) .$ For every $t \neq 0$ this map is not an isometry: there exists $U \in H^{s}$ with $∥ Ψ_{t} (U) ∥_{H^{s - t}} \neq ∥ U ∥_{H^{s}}$ .
Contractive inclusion. If $s \geq r$ , then $κ (U) : = E_{r}^{- 1} (E_{s} U)$ defines an injective complex-linear contraction $κ : H^{s} (R^{n}) \to H^{r} (R^{n})$ with $∥ κ (U) ∥_{H^{r}} \leq ∥ U ∥_{H^{s}}$ for every $U$ ; the weighted class in $H^{r}$ is $⟨ ξ ⟩^{r - s} g$ .
Derivatives lose one order. For every $j \in {1, \dots, n}$ and $U \in H^{s + 1}$ , the distributional derivative $Θ_{j} (U) : = E_{s}^{- 1} (\partial_{j} E_{s + 1} U)$ is well defined in $H^{s}$ , the map $Θ_{j}$ is complex-linear, its weighted class in $H^{s}$ is $2 π i ξ_{j} ⟨ ξ ⟩^{- 1} g_{s + 1}$ , where $g_{s + 1} = ⟨ ξ ⟩^{s + 1} F (E_{s + 1} U) \in L^{2}$ is the weighted Fourier class at order $s + 1$ , and $∥ Θ_{j} (U) ∥_{H^{s}} \leq 2 π ∥ U ∥_{H^{s + 1}} .$

Thus $⟨ D ⟩^{t}$ shifts the Sobolev order exactly and isometrically, while $(I - Δ)^{t / 2}$ shifts it only up to the bounded factor $r_{t}$ , which equals $1$ at $ξ = 0$ and tends to $(2 π)^{t}$ at high frequency; no isometry of $(I - Δ)^{t / 2}$ for the bracket norm is asserted when $t \neq 0$ .

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $s, t \in R$ , the completion $H^{s}$ with canonical embedding $E_{s}$ , and the bracket $⟨ ξ ⟩ \geq 1$ .

[A1]

Countable Choice permits one selection from each nonempty set in a countable family (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

$E_{σ} : H^{σ} \to W_{σ}$ is a bijection onto the tempered distributions $u$ with $⟨ ξ ⟩^{σ} F u = u_{G}$ for a unique $G \in L^{2}$ , and then $∥ U ∥_{H^{σ}} = ∥ G ∥_{2}$ (Real-order H^s as weighted Fourier distributions).

[F2]

$⟨ D ⟩^{t}$ and $(I - Δ)^{t / 2}$ are continuous invertible Fourier multipliers on $S^{'} (R^{n})$ with symbols $⟨ ξ ⟩^{t}$ and $a_{t} (ξ) = (1 + 4 π^{2} ∣ ξ ∣^{2})^{t / 2}$ , and $⟨ D ⟩^{- t}$ , $(I - Δ)^{- t / 2}$ are their inverses (Japanese-bracket and Laplacian Bessel-potential operators).

[F3]

A smooth symbol with every derivative polynomially bounded preserves $S$ and acts on $S^{'}$ by $⟨ a u, φ ⟩ = ⟨ u, a φ ⟩$ (Smooth polynomially bounded multipliers on schwartz space). For the regular distributions used below, write $h = b G$ with $G \in L^{2}$ and $b$ such a smooth multiplier (in particular, a bracket weight or its product with a polynomial or a Laplacian weight). Then $⟨ u_{h}, φ ⟩ = \int G (b φ)$ converges and is continuous by Cauchy–Schwarz and the continuous maps $S \overset{b}{\to} S ↪ L^{2}$ ([F5], Schwartz derivatives are integrable). It agrees on compact tests with the regular functional of Regular distribution from a locally integrable function. For another such multiplier $a$ , $\int G (b a φ) = \int (a h) φ$ , so $a u_{h} = u_{a h}$ . This assertion uses weighted $L^{2}$ densities, not arbitrary locally integrable functions.

[F4]

For every tempered $u$ and multi-index $α$ , $F (D^{α} u) = (2 π i ξ)^{α} F u$ in $S^{'} (R^{n})$ (Distributional derivatives are polynomial Fourier multipliers).

[F5]

Every $L^{2}$ class is locally integrable: for compact $K$ , $\int_{K} ∣ g ∣ \leq ∣ K ∣^{1 / 2} ∥ g ∥_{2} < \infty$ by Cauchy–Schwarz and finiteness of the measure of bounded sets (Complex completeness, density, and inner product: the consumer interface, Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure).

[F6]

$S (R^{n})$ is invariant under $F$ and its inverse (Fourier transform is a topological automorphism of Schwartz space).

[F7]

A smooth compactly supported $χ$ belongs to $S$ , and the inclusion $C_{c}^{\infty} (R^{n}) \to S (R^{n})$ is continuous; furthermore for $0 < r_{1} < r_{2}$ there is a smooth bump equal to $1$ on the closed ball of radius $r_{1}$ and supported in the open ball of radius $r_{2}$ (Test function inclusion in schwartz space is continuous, A Euclidean bump for a compact set inside an open set).

[F8]

$x^{u} = \exp (u \log x)$ for $x > 0$ (Real powers for positive bases, with the zero-base positive-exponent convention); the logarithm satisfies $\log^{'} (x) = 1 / x > 0$ on $(0, \infty)$ and is therefore strictly increasing (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t), and $\exp$ is strictly increasing (The exponential function is strictly increasing), so $x \mapsto x^{u}$ is strictly increasing for $u > 0$ and strictly decreasing for $u < 0$ .

Proof

technique · transport the two Fourier multipliers through the canonical identifications $H^\sigma\cong\mathcal W_\sigma$ and compare the resulting frequency weights

1.1F1F2F3F5given

The bracket transfer. Let $U \in H^{s}$ with class $g = ⟨ ξ ⟩^{s} F (E_{s} U)$ as in [F1] and put $u : = E_{s} U$ , so $F u = u_{⟨ ξ ⟩^{- s} g}$ by [F1] and [F3]. Since $⟨ ξ ⟩^{- s} g$ is locally integrable by [F5], [F3] applied to the smooth symbol $⟨ ξ ⟩^{t}$ and to the symbol $⟨ ξ ⟩^{s - t}$ gives $F (⟨ D ⟩^{t} u) = ⟨ ξ ⟩^{t} F u = u_{⟨ ξ ⟩^{t - s} g}, ⟨ ξ ⟩^{s - t} F (⟨ D ⟩^{t} u) = u_{⟨ ξ ⟩^{s - t} ⟨ ξ ⟩^{t - s} g} = u_{g} .$ Hence $⟨ D ⟩^{t} u \in W_{s - t}$ with weighted class $g$ and, by [F1], $⟨ D ⟩^{t} u = E_{s - t} V$ for a unique $V \in H^{s - t}$ with $∥ V ∥_{H^{s - t}} = ∥ g ∥_{2} = ∥ U ∥_{H^{s}}$ . The assignment $U \mapsto V = Φ_{t} (U)$ is linear because $E_{s}$ , $⟨ D ⟩^{t}$ and $E_{s - t}^{- 1}$ are linear on their domains [F1], [F2]; it preserves the norm and is therefore injective.

1.2F8algebra

The Laplacian weight ratio. Put $ρ (u) = (1 + 4 π^{2} u) / (1 + u)$ for $u \geq 0$ ; then $ρ^{'} (u) = (4 π^{2} - 1) / (1 + u)^{2} > 0$ , so $ρ$ is increasing with $ρ (0) = 1$ and $\lim_{u \to \infty} ρ (u) = 4 π^{2}$ . Writing $r_{t} (ξ) = ρ (∣ ξ ∣^{2})^{t / 2}$ and using [F8], for $t > 0$ the factor $r_{t} (ξ)$ is increasing in $∣ ξ ∣$ with values in $[1, (2 π)^{t}]$ , and for $t < 0$ it is decreasing in $∣ ξ ∣$ with values in $[(2 π)^{t}, 1]$ ; in both cases $c_{t} \leq r_{t} (ξ) \leq C_{t}, c_{t} = \min (1, (2 π)^{t}), C_{t} = \max (1, (2 π)^{t}) .$

1.3F1F3F5given

Contractive inclusion. Let $s \geq r$ and $U \in H^{s}$ with class $g = ⟨ ξ ⟩^{s} F (E_{s} U) \in L^{2}$ . Since $⟨ ξ ⟩^{r - s} \leq 1$ and $g$ is locally integrable [F5], [F3] gives $⟨ ξ ⟩^{r} F (E_{s} U) = u_{⟨ ξ ⟩^{r - s} g}$ , and $⟨ ξ ⟩^{r - s} g \in L^{2}$ with $∥ ⟨ ξ ⟩^{r - s} g ∥_{2} \leq ∥ g ∥_{2}$ . Hence $E_{s} U \in W_{r}$ , so $κ (U) : = E_{r}^{- 1} (E_{s} U)$ is defined, has weighted class $⟨ ξ ⟩^{r - s} g$ , and satisfies $∥ κ (U) ∥_{H^{r}} = ∥ ⟨ ξ ⟩^{r - s} g ∥_{2} \leq ∥ g ∥_{2} = ∥ U ∥_{H^{s}}$ . The map $κ$ is linear by [F1], and it is injective: $κ (U) = 0$ forces $⟨ ξ ⟩^{r - s} g = 0$ almost everywhere and hence $g = 0$ and $U = 0$ .

1.4F1F3F4F5

Derivative bound. Let $U \in H^{s + 1}$ with weighted class $g_{s + 1} = ⟨ ξ ⟩^{s + 1} F (E_{s + 1} U)$ and $u : = E_{s + 1} U$ , so $F u = u_{⟨ ξ ⟩^{- (s + 1)} g_{s + 1}}$ and $∥ U ∥_{H^{s + 1}} = ∥ g_{s + 1} ∥_{2}$ [F1]. By [F4], $F (\partial_{j} u) = 2 π i ξ_{j} F u$ , and $⟨ ξ ⟩^{- (s + 1)} g_{s + 1}$ is locally integrable [F5], so [F3] gives $F (\partial_{j} u) = u_{2 π i ξ_{j} ⟨ ξ ⟩^{- (s + 1)} g_{s + 1}}, ⟨ ξ ⟩^{s} F (\partial_{j} u) = u_{2 π i ξ_{j} ⟨ ξ ⟩^{- 1} g_{s + 1}} .$ Since $∣ 2 π ξ_{j} ⟨ ξ ⟩^{- 1} ∣ \leq 2 π$ , the class $2 π i ξ_{j} ⟨ ξ ⟩^{- 1} g_{s + 1}$ lies in $L^{2}$ with $∥ 2 π i ξ_{j} ⟨ ξ ⟩^{- 1} g_{s + 1} ∥_{2} \leq 2 π ∥ g_{s + 1} ∥_{2}$ . Hence $\partial_{j} u \in W_{s}$ , so $Θ_{j} (U) : = E_{s}^{- 1} (\partial_{j} E_{s + 1} U)$ is well defined, linear by [F1] and [F4], and $∥ Θ_{j} (U) ∥_{H^{s}} = ∥ 2 π i ξ_{j} ⟨ ξ ⟩^{- 1} g_{s + 1} ∥_{2} \leq 2 π ∥ g_{s + 1} ∥_{2} = 2 π ∥ U ∥_{H^{s + 1}}$ , with weighted class $2 π i ξ_{j} ⟨ ξ ⟩^{- 1} g_{s + 1}$ .

2.1F1F2F3F5step 1.1

Surjectivity and inverse of the bracket shift. Let $V \in H^{s - t}$ with class $h = ⟨ ξ ⟩^{s - t} F (E_{s - t} V)$ , and set $g : = h$ ; because $h$ is locally integrable by [F5], the distribution $u : = F^{- 1} (u_{⟨ ξ ⟩^{- s} h})$ satisfies $⟨ ξ ⟩^{s} F u = u_{h}$ , so $u \in W_{s}$ and $u = E_{s} U$ for some $U \in H^{s}$ with class $g = h$ and $∥ U ∥_{H^{s}} = ∥ h ∥_{2} = ∥ V ∥_{H^{s - t}}$ [F1]. Since $F (⟨ D ⟩^{t} u) = u_{⟨ ξ ⟩^{t - s} h}$ and $F (E_{s - t} V) = u_{⟨ ξ ⟩^{t - s} h}$ by [F2], [F1] and [F3], injectivity of $F$ on $S^{'}$ [F2] gives $⟨ D ⟩^{t} E_{s} U = E_{s - t} V$ , that is, $Φ_{t} (U) = V$ . Hence $Φ_{t}$ is surjective, and step 1.1 makes it additive, so $Φ_{t}$ is a complex-linear bijection. Finally $Φ_{- t} (Φ_{t} (U)) = E_{s}^{- 1} ⟨ D ⟩^{- t} E_{s - t} E_{s - t}^{- 1} ⟨ D ⟩^{t} E_{s} U = E_{s}^{- 1} E_{s} U = U$ by the inverse laws [F2], so $Φ_{- t}$ is the inverse of $Φ_{t}$ .

2.2F1F2F3F5step 1.1step 1.2

The Laplacian transfer. Let $U \in H^{s}$ with class $g$ and $u = E_{s} U$ as in step 1.1. Since $⟨ ξ ⟩^{- s} g$ is locally integrable [F5], [F3] gives $F ((I - Δ)^{t / 2} u) = a_{t} F u = u_{a_{t} ⟨ ξ ⟩^{- s} g}, ⟨ ξ ⟩^{s - t} F ((I - Δ)^{t / 2} u) = u_{r_{t} g},$ using $a_{t} ⟨ ξ ⟩^{s - t} ⟨ ξ ⟩^{- s} = ⟨ ξ ⟩^{- t} a_{t} = r_{t}$ . Hence $(I - Δ)^{t / 2} u \in W_{s - t}$ with class $r_{t} g$ , so $Ψ_{t} (U) : = E_{s - t}^{- 1} ((I - Δ)^{t / 2} E_{s} U)$ is well defined, linear by [F1] and [F2], and by [F1] and step 1.2 $c_{t} ∥ U ∥_{H^{s}} = c_{t} ∥ g ∥_{2} \leq ∥ r_{t} g ∥_{2} = ∥ Ψ_{t} (U) ∥_{H^{s - t}} \leq C_{t} ∥ g ∥_{2} = C_{t} ∥ U ∥_{H^{s}} .$ Conversely, for $V \in H^{s - t}$ with class $h$ put $g : = r_{t}^{- 1} h$ ; step 1.2 gives $∣ g ∣ \leq c_{t}^{- 1} ∣ h ∣$ , so $g \in L^{2}$ , and $u : = F^{- 1} (u_{⟨ ξ ⟩^{- s} g})$ lies in $W_{s}$ [F5], [F3]; then $⟨ ξ ⟩^{s - t} F ((I - Δ)^{t / 2} u) = u_{r_{t} g} = u_{h}$ , so $(I - Δ)^{t / 2} u = E_{s - t} V$ and $Ψ_{t}$ is onto. Thus $Ψ_{t}$ is a bounded complex-linear bijection with the displayed two-sided bounds.

3.1F1F3F6F7step 1.2step 2.2

The Laplacian shift is not isometric for $t \neq 0$ . Fix $t \neq 0$ and choose $R \geq 1$ with $r_{t} (ξ) > 1$ for all $∣ ξ ∣ \geq R$ if $t > 0$ , respectively $r_{t} (ξ) < 1$ for all $∣ ξ ∣ \geq R$ if $t < 0$ ; this is possible because $r_{t} (ξ) \to (2 π)^{t}$ as $∣ ξ ∣ \to \infty$ and $(2 π)^{t} > 1$ for $t > 0$ , $(2 π)^{t} < 1$ for $t < 0$ , while $r_{t}$ is monotone in $∣ ξ ∣$ by step 1.2. By [F7] with $K = {2 R e_{1}}$ and the open ball $V = B (2 R e_{1}, R / 2) \subseteq {∣ ξ ∣ > R}$ , there is a smooth bump $χ$ equal to $1$ on $K$ and supported in $V$ . Thus $χ \neq 0$ , and $χ \in S$ by [F7]. Set $φ : = F^{- 1} χ \in S$ [F6] and $G : = ⟨ ξ ⟩^{s} χ$ , which lies in $L^{2}$ because $χ$ has compact support and $⟨ ξ ⟩^{s}$ is bounded there; let $U : = E_{s}^{- 1} (u_{φ})$ be the canonical class of $φ$ , whose weighted class is $G$ by [F1], [F3]: $∥ U ∥_{H^{s}} = ∥ G ∥_{2} > 0$ . By step 2.2 the class of $Ψ_{t} (U)$ is $r_{t} G$ , so $∥ Ψ_{t} (U) ∥_{H^{s - t}}^{2} = \int ∣ χ ∣^{2} r_{t}^{2} ⟨ ξ ⟩^{2 s} d ξ {\begin{cases} > \\ < \end{cases} \int ∣ χ ∣^{2} ⟨ ξ ⟩^{2 s} d ξ = ∥ U ∥_{H^{s}}^{2},$ with the strict inequality $>$ when $t > 0$ and $<$ when $t < 0$ , because $r_{t}^{2} > 1$ respectively $r_{t}^{2} < 1$ on $supp χ$ and $χ \neq 0$ there. Hence $Ψ_{t}$ is not an isometry for any $t \neq 0$ .

4.1A1F1F2F3F4F5F6F7F8step 1.1step 2.1step 1.2step 2.2step 3.1step 1.3step 1.4∎

Conclusion. Step 1.1 and step 2.1 give the surjective isometry $Φ_{t} : H^{s} \to H^{s - t}$ with inverse $Φ_{- t}$ ; step 1.2 and step 2.2 give the bounded bijection $Ψ_{t}$ with the two-sided bounds; step 3.1 produces, for every $t \neq 0$ , an explicit frequency-localized witness showing that $Ψ_{t}$ is not an isometry; step 1.3 gives the contractive inclusion for $s \geq r$ ; and step 1.4 gives the derivative bound with constant $2 π$ . This proves statements 1-4 for arbitrary $n \geq 1$ and $s, t \in R$ . Countable Choice is used exactly through the cited completion and characterization interfaces, which carry it as their hypothesis.

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Sources

Semyon Dyatlov, Lecture Notes for 18.155, current revision (standard reference, not scraped)
Mark Williams, Notes on Harmonic Analysis (standard reference, not scraped)