Optimal regularity results in spaces of Hölder continuous functions for some infinite dimensional Ornstein–Uhlenbeck semigroup

Abstract

We consider the elliptic equation $\lambda \phi –L\phi =f$ where $\lambda >0$, $f$ is $\theta$-Hölder continuous and $L$ is an Ornstein−Uhlenbeck operator in a Hilbert space $H$. We show that the mapping $D^2\phi$ (with values in the space of Hilbert−Schmidt operators on $H$) is $\theta$-Hölder continuous.