Multiplier theorems via martingale transforms

摘要

We develop a new and general approach to prove multiplier theorems in various geometric settings. The main idea is to use martingale transforms and a Gundy-Varopoulos representation for multipliers defined via a suitable extension procedure. Along the way, we provide a probabilistic proof of a generalization of a result by Stinga and Torrea, which is of independent interest. Our methods here also recover the sharp Lpbounds for second order Riesz transforms by a limiting argument. (C) 2021 Elsevier Inc. All rights reserved.