On Some Operators Associated with Non-Degenerate Symmetric α-Stable Probability Measures

Abstract

Boundedness properties of operators associated with non-degenerate symmetric α-stable, α ∈ (1,2), probability measures on \(\mathbb {R}^{d}\) are investigated on appropriate, Euclidean or otherwise, Lp-spaces, \(p \in (1,+\infty )\). Our approach is based on firstly obtaining Bismut-type formulas 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 α-stable probability measure.