Optimal Sobolev embeddings for the Ornstein-Uhlenbeck operator
Abstract
A comprehensive analysis of Sobolev-type inequalities for the Ornstein-Uhlenbeck operator in the Gauss space is offered. A unified approach is proposed, providing one with criteria for their validity in the class of rearrangement-invariant function norms. Optimal target and domain norms in the relevant inequalities are characterized via a reduction principle to one-dimensional inequalities for a Calder\'on type integral operator patterned on the Gaussian isoperimetric function. Consequently, the best possible norms in a variety of specific families of spaces, including Lebesgue, Lorentz, Lorentz-Zygmund, Orlicz and Marcinkiewicz spaces, are detected. The reduction principle hinges on a preliminary discussion of the existence and uniqueness of generalized solutions to equations, in the Gauss space, for the Ornstein-Uhlenbeck operator, with a just integrable right-hand side. A decisive role is also played by a pointwise estimate, in rearrangement form, for these solutions.
Cite
@article{arxiv.2209.14193,
title = {Optimal Sobolev embeddings for the Ornstein-Uhlenbeck operator},
author = {Andrea Cianchi and Vít Musil and Luboš Pick},
journal= {arXiv preprint arXiv:2209.14193},
year = {2023}
}