Endpoint bounds for quasiradial Fourier multipliers

TL;DR

This paper establishes necessary and sufficient conditions for the boundedness of quasiradial Fourier multipliers and their maximal operators on L^p spaces, advancing understanding of their endpoint behavior.

Contribution

It provides a complete characterization of endpoint bounds for quasiradial Fourier multipliers and associated maximal operators, which was previously unknown.

Findings

Characterization of boundedness conditions for quasiradial Fourier multipliers

Results on maximal operators when the multiplier is compactly supported

Extension of bounds to endpoint cases for L^p spaces

Abstract

We consider quasiradial Fourier multipliers, i.e. multipliers of the form $m(a(\xi))$ for a class of distance functions $a$. We give a necessary and sufficient condition for the multiplier transformations to be bounded on $L^p$ for a certain range of $p$. In addition, when $m$ is compactly supported in $(0,\infty)$, we give a similar result for associated maximal operators.