On Some Operators Associated with Non-Degenerate Symmetric $\hat{I}\pm$-Stable Probability Measures

Abstract

Boundedness properties of operators associated with non-degenerate symmetric $\hat{I}\pm$-stable, $\hat{I}\pm \in (1,2)$, probability measures on ${\mathbb{R}}^d$ are investigated on appropriate, Euclidean or otherwise, $L^p$-spaces, $p\in (1,+\infty )$. Our approach is based on first obtaining Bismut-type formulae which lead to useful representations for various operators. In the Euclidean setting, the method of transference and one-dimensional multiplier theory combined with fine properties of stable distributions provide dimension-free estimates for the fractional Laplacian. In the non-Euclidean setting, we obtain boundedness results for the non-singular cases as well as dimension-free estimates when the reference measure is the rotationally invariant $\hat{I}\pm$-stable probability measure.

53 pages