On the equivalence of Sobolev norms in infinite dimensions

Abstract

We prove dimension-free higher-order Sobolev norm estimates on open convex subsets of ${\mathbb{R}}^n$ with respect to Gaussian measure and use them to obtain norm equivalence on nonempty open convex subsets of ${\ell }^2$ endowed with a nondegenerate Gaussian measure. To the best of our knowledge, this is the first such equivalence theorem on a proper open subset of an infinite-dimensional Hilbert space, beyond the earlier whole-space results. We also prove the Malliavin--Sobolev norm equivalence for all $p\in [1,\infty )$ and $k\ge 2$, including the case $p=1$, $k\ge 3$ left open by Addona--Muratori--Rossi in \cite{AddonaMuratoriRossi}.