Martingale transform and L\'evy Processes on Lie Groups

TL;DR

This paper develops a framework for martingale transforms based on Lévy processes on Lie groups, leading to new bounded operators on L^p spaces, including Fourier multipliers like Riesz transforms and imaginary powers of the Laplacian.

Contribution

It introduces a novel class of martingale transforms on Lie groups that produce bounded operators on L^p spaces, extending previous Euclidean results to the Lie group setting.

Findings

Constructed martingale transforms from Lévy processes on Lie groups.

Derived bounded Fourier multipliers including Riesz transforms and imaginary powers.

Extended and simplified previous Euclidean results to Lie groups.

Abstract

This paper constructs a class of martingale transforms based on L\'evy processes on Lie groups. From these, a natural class of bounded linear operators on the $L^p$-spaces of the group (with respect to Haar measure) for $1<p<\infty$, are derived. On compact groups these operators yield Fourier multipliers (in the Peter-Weyl sense) which include the second order Riesz transforms, imaginary powers of the Laplacian, and new classes of multipliers obtained by taking the L\'evy process to have conjugate invariant laws. Multipliers associated to subordination of the Brownian motion on the group are special cases of this last class. These results extend (and the proofs simplify) those obtained in \cite{BanBieBog, BanBog} for the case of $\bR^n$. An important feature of this work is the optimal nature of the $L^p$ bounds.