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Distributional derivatives are polynomial Fourier multipliers

Statement

Assume Countable Choice and let $n \geq 1$ . For every $u \in S^{'} (R^{n})$ and every multi-index $α \in N_{0}^{n}$ , $F (D^{α} u) = (2 π i ξ)^{α} F u in S^{'} (R^{n}),$ where $D^{α}$ is the distributional derivative of $C^{k}$ maps and multi-index derivative notation in Euclidean space and $F$ is the negative-sign $2 π$ -normalized transform. Moreover, if $u = u_{f}$ and $D^{α} u = u_{g}$ for $L^{2}$ classes $f, g \in L^{2} (R^{n})$ and their regular distributions, then the unitary Plancherel transforms satisfy $F_{2} g (ξ) = (2 π i ξ)^{α} F_{2} f (ξ) for almost every ξ \in R^{n} .$ The second assertion compares the Plancherel classes only; it neither asserts pointwise values of arbitrary representatives nor presupposes any Sobolev-space notation.

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $u \in S^{'} (R^{n})$ , a multi-index $α$ , and, for the second assertion, $L^{2}$ classes $f, g$ with $u = u_{f}$ and $D^{α} u = u_{g}$ .

[A1]

Countable Choice is the hypothesis carried by the cited tempered distribution and Plancherel interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The distributional derivative of a tempered distribution is tempered and $F (\partial^{α} u) = (2 π i ξ)^{α} F u$ in $S^{'}$ , with the conventions $⟨ \partial^{α} u, φ ⟩ = (- 1)^{∣ α ∣} ⟨ u, \partial^{α} φ ⟩$ and bilinear test pairing (Fourier differentiation and multiplication identities on tempered distributions).

[F2]

Every complex $L^{p}$ class, $1 \leq p \leq \infty$ , has a representative whose regular distribution is tempered; in particular $L^{2}$ classes define tempered distributions (Polynomial growth functions define tempered distributions).

[F3]

The locally integrable regular distribution is $u_{h} (φ) = \int h φ$ , with bilinear pairing, and the map $h \mapsto u_{h}$ factors through almost-everywhere equality (Regular distribution from a locally integrable function).

[F4]

The regular-distribution map on locally integrable functions is injective after almost-everywhere identification (Locally integrable functions embed in distributions).

[F5]

For $h \in L^{2} (R^{n})$ the distributional transform of the regular distribution is the regular distribution of the Plancherel transform: $F u_{h} = u_{F_{2} h}$ in $S^{'}$ (Fourier transform agrees with l one and plancherel transforms).

[F6]

Multiplication of a tempered distribution $w$ by a smooth function $a$ with polynomially bounded derivatives is the tempered distribution $⟨ a w, φ ⟩ = ⟨ w, a φ ⟩$ (Smooth polynomially bounded multipliers on schwartz space).

[F7]

Plancherel extends the Schwartz transform to a surjective complex-linear isometry $F_{2} : L^{2} \to L^{2}$ (Plancherel theorem).

Proof

technique · transpose the published differentiation identity, then compare regular distributions

1.1F1

The multi-index derivative $D^{α} u$ is the iterated distributional partial derivative, so [F1] applies verbatim and gives $F (D^{α} u) = (2 π i ξ)^{α} F u$ in $S^{'}$ ; this includes $α = 0$ , where the multiplier is the constant $1$ .

1.2F2F5

Assume now that $u = u_{f}$ and $D^{α} u = u_{g}$ for $L^{2}$ classes $f, g$ . Both regular distributions are tempered by [F2], and [F5] identifies their transforms as $F u_{f} = u_{F_{2} f}$ and $F u_{g} = u_{F_{2} g}$ .

2.1F3F6step 1.1step 1.2

Substituting step 1.2 into step 1.1 applied to $u_{f}$ gives $u_{F_{2} g} = (2 π i ξ)^{α} u_{F_{2} f} = u_{(2 π i ξ)^{α} F_{2} f}$ : the last equality follows from the product rule [F6] with the smooth polynomially bounded multiplier $(2 π i ξ)^{α}$ together with the defining formula [F3], since both sides pair a test $φ$ with $\int_{R^{n}} (2 π i ξ)^{α} F_{2} f (ξ) φ (ξ) d ξ$ .

3.1F3F4F7step 2.1

Since $F_{2}$ is a surjective isometry of $L^{2}$ [F7], the class $F_{2} f$ lies in $L^{2}$ ; the function $ξ \mapsto (2 π i ξ)^{α} F_{2} f (ξ)$ is a polynomially growing multiple of it and hence is locally integrable, so both sides of step 2.1 are regular distributions of locally integrable functions; injectivity of that map [F4] yields $F_{2} g = (2 π i ξ)^{α} F_{2} f$ almost everywhere.

4.1A1step 1.1step 3.1∎

Countable Choice is used only through the cited tempered-distribution and Plancherel interfaces [A1]; the transposition computation of step 1.1 and the injectivity argument of step 3.1 add no further choice.

Sources

Semyon Dyatlov, Lecture Notes for 18.155, §12.1.1, Proposition 12.1 and proof, printed pp. 139-140. The source uses $D = - i \partial$ and unit normalization; its identity is converted here to the repository convention $F (\partial_{j} u) = 2 π i ξ_{j} F u$ .
Mark Williams, Notes on Harmonic Analysis, §5.3 and §6.2, printed pp. 19-25, for the multiplier and Sobolev conventions in which the polynomial symbol is consumed.

Depends on

Used by

Translation and differentiation symbols Example
Bessel potentials shift Sobolev order Lemma
Integer-order W^k,2 and Hᵏ agree with equivalent norms Theorem

Dependency tree · two levels

61 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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