A Method of Rotations for L\'evy Multipliers

TL;DR

This paper introduces a rotation-based method to analyze the $L^p$ boundedness of Fourier multipliers linked to symmetric $eta$-stable processes, expanding the class of multipliers known to be bounded on $L^p$ spaces.

Contribution

It presents a novel rotation method that does not depend on the stability parameter, allowing for broader $L^p$ boundedness results for certain Fourier multipliers.

Findings

Established $L^p$ boundedness for a larger class of multipliers

Provided a new approach independent of the stability parameter

Potential applications to the Beurling-Ahlfors transform

Abstract

We use a method of rotations to study the $L^p$ boundedness, $1<p<\infty$, of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric $\alpha$-stable processes, $0<\alpha<2$. Our proof does not use the fact that $0<\alpha<2$, and therefore allows us to obtain a larger class of multipliers which are bounded on $L^p$. As in the case of the multipliers which arise as the projection of martingale transforms, these new multipliers also have potential applications to the study of the $L^p$ boundedness of the Beurling-Ahlfors transform.