Dimension Free Estimates for the Riesz Transforms Associated with some Fractional Operators

Benjamin Arras, Christian Houdré

Published 2020-05-13Version 1

In these notes, boundedness properties of the Riesz transforms associated with symmetric $\alpha$-stable probability measures, $\alpha \in (1,2)$, are investigated on appropriate $L^p$ spaces, for $p \in (1,+\infty)$. Our approach is based on Bismut-type formulae in order to obtain useful representations for the different Riesz transforms. In the Euclidean setting, the method of transference and one-dimensional multiplier theory combined with fine properties of stable distributions imply dimension free estimate for the fractional Laplacian. A limiting argument as $\alpha$ tends to $2^-$ then recovers the well-known result for the standard Laplacian. In the non-Euclidean setting, a regularization phenomenon, specific to the non-Gaussian stable case, provides the boundedness result as well as a dimension free estimate when the reference measure is the rotationally invariant $\alpha$-stable probability measure on $\mathbb{R}^d$.