Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a theorem of M. Jodeit Jr. on pushforwards of Fourier multipliers

Patrick Poissel

Abstract

A classical theorem of M. Jodeit Jr. implies that if a compactly supported distribution on $\mathbf{R}^d$ is the symbol of a Fourier multiplier bounded from $L^p(\mathbf{R}^d)$ to $L^q(\mathbf{R}^d)$, then its pushforward by the canonical homomorphism from $\mathbf{R}^d$ to $\mathbf{T}^d$ is the symbol of a Fourier multiplier bounded from $\ell^p(\mathbf{Z}^d)$ to $\ell^q(\mathbf{Z}^d)$. In the present work, we generalise this result to the setting of locally compact groups, including those non-abelian, by characterising the continuous homomorphisms of locally compact groups by which, for every $p,q\in[1,\infty]$, the pushforward of compactly supported distributions symbols of $L^p$-$L^q$ Fourier multipliers are symbols of the same type as those which are open. Motivated by a simple proof in the abelian case, we also investigate pushforwards of positive definite distributions.

On a theorem of M. Jodeit Jr. on pushforwards of Fourier multipliers

Abstract

A classical theorem of M. Jodeit Jr. implies that if a compactly supported distribution on is the symbol of a Fourier multiplier bounded from to , then its pushforward by the canonical homomorphism from to is the symbol of a Fourier multiplier bounded from to . In the present work, we generalise this result to the setting of locally compact groups, including those non-abelian, by characterising the continuous homomorphisms of locally compact groups by which, for every , the pushforward of compactly supported distributions symbols of - Fourier multipliers are symbols of the same type as those which are open. Motivated by a simple proof in the abelian case, we also investigate pushforwards of positive definite distributions.
Paper Structure (18 sections, 41 theorems, 128 equations)

This paper contains 18 sections, 41 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Terminology and notations
  3. Integration
  4. Distributions
  5. Pushforward and pullback operators
  6. Generalities on Fourier multipliers
  7. Classical Fourier multipliers
  8. Lebesgue spaces on the dual of a locally compact group
  9. Non-commutative Fourier multipliers
  10. The general pushforward theorem
  11. Reduction to the case p=q=1
  12. Schur multipliers lemmas
  13. Proof of the general pushforward theorem
  14. Functorial properties of positive definite distributions
  15. Positive definite distributions and amenability
  16. ...and 3 more sections

Key Result

theorem 1.0

Let $G$ and $H$ be two abelian locally compact groups and let $\pi$ be a continuous homomorphism from $G$ to $H$. Let $m$ be a sufficiently regular complex function on $H$ and let $p\in[1,\infty]$. For $m\circ\pi$ to be the symbol of Fourier multiplier bounded on $L^p(\widehat{G{}})$, it is sufficie If additionally $m$ is supported in the closure of $\pi(G)$ in $H$, then for $m\circ\pi$ to be the

Theorems & Definitions (47)

  • theorem 1.0: N. Lohoué LohoueEnssyntheseLohoueApprox
  • theorem 1.0: M. Jodeit Jr. Jodeit
  • corollary 1.0
  • theorem 1.0
  • proposition 2.0
  • proposition 3.0
  • proposition 3.0
  • proposition 3.0
  • proposition 3.0
  • lemma 3.0
  • ...and 37 more