On the equivalence of Sobolev norms in Malliavin spaces

TL;DR

This paper studies the equivalence of Sobolev norms in Malliavin spaces, establishing new results for the challenging case q=1 and providing explicit bounds in both finite and infinite-dimensional settings.

Contribution

It proves the equivalence of Sobolev norms for q=1 and k=2 in Malliavin spaces using a novel vector-valued Poincaré inequality, and offers explicit bounds in finite and infinite dimensions.

Findings

Established Sobolev norm equivalence for q=1, k=2 in infinite-dimensional Malliavin spaces.

Provided explicit bounds on constants in Sobolev inequalities for finite-dimensional Gaussian spaces.

Extended known results for q in (1, ∞) with more direct and quantitative proofs.

Abstract

We investigate the problem of the equivalence of $L^q$-Sobolev norms in Malliavin spaces for $q\in [1,\infty)$, focusing on the graph norm of the $k$-th Malliavin derivative operator and the full Sobolev norm involving all derivatives up to order $k$, where $k$ is any positive integer. The case $q=1$ in the infinite-dimensional setting is challenging, since at such extreme the standard approach involving Meyer's inequalities fails. In this direction, we are able to establish the mentioned equivalence for $q=1$ and $k=2$ relying on a vector-valued Poincar\'e inequality that we prove independently and that turns out to be new at this level of generality, while for $q=1$ and $k>2$ the equivalence issue remains open, even if we obtain some functional estimates of independent interest. With our argument (that also resorts to the Wiener chaos) we are able to recover the case $q\in (1,\infty)$ in the infinite-dimensional setting; the latter is known since the eighties, however our proof is more direct than those existing in the literature, and allows to give explicit bounds on all the multiplying constants involved in the functional inequalities. Finally, we also deal with the finite-dimensional case for all $q\in [1,\infty)$ (where the equivalence, without explicit constants, follows from standard compactness arguments): our proof in such setting is much simpler, relying on Gaussian integration-by-parts formulas and an adaptation of Sobolev inequalities in Euclidean spaces, and it provides again quantitative bounds on the multiplying constants, which however blow up when the dimension diverges to $\infty$ (whence the need for a different approach in the infinite-dimensional setting).