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The Fourier transform is an extremizer of a class of bounded operators

Miquel Saucedo, Sergey Tikhonov

Abstract

We show that, for a natural class of rearrangement admissible spaces $X$ and $Y$, the Fourier operator is bounded between $X$ and $Y$ if and only if any operator of joint strong type $(1,\infty; 2,2)$ is also bounded between $X$ and $Y$. By using this result, we fully characterize the weighted Fourier inequalities of the form $$\qquad\qquad \lVert\widehat{f}u \rVert_q \leq C \lVert fv\rVert_p,\quad 1\leq p\leq \infty,\,0<q\leq \infty,$$ for radially monotone weights $(u,v)$. This answers a long-standing problem posed by Benedetto-Heinig, Jurkat-Sampson, and Muckenhoupt. In the case of $p\le q$, such a characterization has been known since the 1980s.

The Fourier transform is an extremizer of a class of bounded operators

Abstract

We show that, for a natural class of rearrangement admissible spaces and , the Fourier operator is bounded between and if and only if any operator of joint strong type is also bounded between and . By using this result, we fully characterize the weighted Fourier inequalities of the form for radially monotone weights . This answers a long-standing problem posed by Benedetto-Heinig, Jurkat-Sampson, and Muckenhoupt. In the case of , such a characterization has been known since the 1980s.

Paper Structure

This paper contains 26 sections, 28 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of the problem
  4. Notation
  5. Main results
  6. Applications
  7. Structure of the article
  8. Auxiliary results
  9. The method of Calderón
  10. Duality
  11. Khinchin's inequality and factorization
  12. Hardy's and related inequalities
  13. Necessary conditions for weighted Fourier inequalities
  14. The case p>2
  15. The case q<p
  16. ...and 11 more sections

Key Result

Theorem 1.5

Let $\left\lVert\cdot\right\rVert_X$ and $\left\lVert\cdot\right\rVert_Y$ be left- and right-admissible, respectively, and $0< \beta\leq 1$. Then, the following are equivalent$:$ Moreover, the optimal constants satisfy $C_1 \approx_{\beta} C_2$.

Theorems & Definitions (56)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Definition 2.1
  • Lemma 2.2: bennett1988interpolation and jodeit
  • proof
  • ...and 46 more