Martingale transforms and the Hardy-Littlewood-Sobolev inequality for semigroups

TL;DR

This paper presents a novel representation of fractional integrals for symmetric Markovian semigroups using martingale transforms and establishes the Hardy-Littlewood-Sobolev inequality through this framework, supported by a new fractional Littlewood-Paley inequality.

Contribution

It introduces a new martingale transform representation for fractional integrals and proves the HLS inequality based on this approach.

Findings

Representation of fractional integrals as martingale transforms

Proof of Hardy-Littlewood-Sobolev inequality using this representation

Development of a new fractional Littlewood-Paley g-function inequality

Abstract

We give a representation of the fractional integral for symmetric Markovian semigroups as the projection of martingale transforms and prove the Hardy-Littlewood-Sobolev(HLS) inequality based on this representation. The proof rests on a new inequality for a fractional Littlewood-Paley $g$-function.