Martingales and Sharp Bounds for Fourier multipliers

TL;DR

This paper employs martingale techniques and a new inequality to precisely determine the $L^p$--norms of specific Fourier multipliers in higher dimensions, including second order Riesz transforms and Lévy multipliers.

Contribution

It introduces a novel martingale inequality and applies it to identify sharp $L^p$ bounds for certain Fourier multipliers in multiple dimensions.

Findings

Exact $L^p$--norms for second order Riesz transforms determined.

Sharp bounds established for Lévy multipliers.

Extension of martingale methods to Fourier analysis in higher dimensions.

Abstract

Using the argument of Geiss, Montgomery-Smith and Saksman \cite{GMSS}, and a new martingale inequality, the $L^p$--norms of certain Fourier multipliers in $\R^d$, $d\geq 2$, are identified. These include, among others, the second order Riesz transforms $R_j^2$, $j=1, 2,..., d$, and some of the L\'evy multipliers studied in \cite{BBB}, \cite{BB}