The Fourier transform in weighted rearrangement invariant spaces

TL;DR

This paper characterizes when the Fourier transform is bounded on weighted rearrangement-invariant spaces, showing it implies the space is essentially $L^2$ with bounded weights, and explores related boundedness conditions.

Contribution

It establishes necessary conditions for the boundedness of the Fourier transform on weighted rearrangement-invariant spaces, linking it to $L^2$ and specific $L^p$-$L^q$ bounds, and applies these results to Schrödinger-related operators.

Findings

Fourier transform boundedness implies space is equivalent to $L^2$ with bounded weights.

Boundedness from $L^p$ to $L^q$ requires $1 q p=q'q 2$ with bounded weights.

Certain Schrödinger-related operators are not bounded on these spaces.

Abstract

It is shown that if the Fourier transform is a bounded map on a rearrangement-invariant space of functions on $\mathbb R^n$, modified by a weight, then the weight is bounded above and below and the space is equivalent to $L^2$. Also, if it is bounded from $L^p$ to $L^q$, each modified by the same weight, then the weight is bounded above and below and $1\le p=q'\le 2$. Applications prove the non-boundedness on these spaces of an operator related to the Schr\"odinger equation.