Malliavin smoothness on the L\'evy space with H\"older continuous or $BV$ functionals

We consider Malliavin smoothness of random variables $f(X_1)$, where $X$ is a pure jump L'evy process and $f$ is either bounded and H"older 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 L'evy measure.