Martingales and Sharp Bounds for Fourier multipliers

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 ${\mathbb{R}}^d$, $d\ge 2$, are identified. These include, among others, the second order Riesz transforms $R_j^2$, $j=1,2,...,d$, and some of the Lévy multipliers studied in \cite{BBB}, \cite{BB}