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We consider Malliavin smoothness of random variables $f(X_1)$, where $X$ is apure jump L\'evy process and $f$ is either bounded and H\"older continuous orof bounded variation. We show that Malliavin differentiability and fractionaldifferentiability of $f(X_1)$ depend both on the regularity of $f$ and theBlumenthal-Getoor index of the L\'evy measure.

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