Optimal Sobolev embeddings for the Ornstein-Uhlenbeck operator

TL;DR

This paper establishes optimal Sobolev inequalities for the Ornstein-Uhlenbeck operator in Gaussian spaces, characterizing the best norms across various function spaces through a reduction to one-dimensional inequalities.

Contribution

It introduces a unified approach to Sobolev inequalities for the Ornstein-Uhlenbeck operator, identifying optimal norms in multiple function spaces via a reduction principle and detailed analysis.

Findings

Characterization of optimal target and domain norms in Sobolev inequalities.

Reduction of multidimensional inequalities to one-dimensional Calderón type operators.

Existence and uniqueness results for solutions to Ornstein-Uhlenbeck equations in Gauss space.

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.