Martingale Transforms and the Hardy-Littlewood-Sobolev Inequality for Semigroups

摘要

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 the fractional Littlewood-Paley g-function.