Hardy-Stein identities and square functions for semigroups

TL;DR

This paper establishes Hardy-Stein identities for symmetric pure-jump Lévy process semigroups, leading to new $L^p$ boundedness results for associated square functions and Fourier multipliers, enhancing understanding of their harmonic analysis properties.

Contribution

It introduces Hardy-Stein identities for Lévy process semigroups and uses them to prove $L^p$ boundedness of square functions and Fourier multipliers, providing new tools for harmonic analysis.

Findings

Hardy-Stein identities for symmetric pure-jump Lévy semigroups

$L^p$ boundedness of Littlewood-Paley square functions

Boundedness of Fourier multipliers from Lévy process martingales

Abstract

We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump L\'evy processes. Combined with the Burkholder-Gundy inequalities, it gives the $L^p$ two-way boundedness, for $1<p<\infty$, of the corresponding Littlewood-Paley square function. The square function yields a direct proof of the $L^p$ boundedness of Fourier multipliers obtained by transforms of martingales of L\'evy processes.