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arXiv · 2209.14193

Optimal Sobolev embeddings for the Ornstein-Uhlenbeck operator

Andrea Cianchi·Vít Musil·Luboš Pick
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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.

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Andrea Cianchi, Vít Musil, Luboš Pick. 2022-09-28. Optimal Sobolev embeddings for the Ornstein-Uhlenbeck operator. https://doi.org/10.1016/j.jde.2023.02.035

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