On some smoothening effects of the transition semigroup of a Lévy process

Let $(P_t)$ be the transition semigroup of a Lévy process $L$ taking values in a Hilbert space $H$. Let $ν$ be the Lévy measure of $L$. It is shown that for any bounded and measurable function $f$, $$ \int_H\left\vert P_tf(x+y)-P_tf(x)\right\vert ^2 ν(\dif y)\le \frac 1 t P_tf^2(x) \qquad \text{for all $t>0$, $x\in H$.} $$ As $ν$ can be infinite this formula establishes some smoothening effect of the semigroup $(P_t)$. In the paper some applications of the formula will be presented as well.

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