Malliavin smoothness on the Lévy space with Hölder continuous or $BV$ functionals

Abstract

We consider Malliavin smoothness of random variables $f(X_1)$, where $X$ is a pure jump Lévy process and $f$ is either bounded and Hölder continuous or of bounded variation. We show that Malliavin differentiability and fractional differentiability of $f(X_1)$ depend both on the regularity of $f$ and the Blumenthal-Getoor index of the Lévy measure.