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arXiv · 2302.07047

The Fourier transform in weighted rearrangement invariant spaces

Mieczysław Mastyło·Gord Sinnamon
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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.

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Mieczysław Mastyło, Gord Sinnamon. 2023-02-14. The Fourier transform in weighted rearrangement invariant spaces. https://doi.org/10.1090/proc/16610

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