Optimal Sobolev embeddings for the Ornstein-Uhlenbeck operator
| Authors | |
|---|---|
| Year of publication | 2023 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Differential Equations |
| MU Faculty or unit | |
| Citation | |
| Web | https://www.sciencedirect.com/science/article/pii/S0022039623001110 |
| Doi | http://dx.doi.org/10.1016/j.jde.2023.02.035 |
| Keywords | Ornstein-Uhlenbeck operator; Gauss space; embeddings; optimality |
| Attached files | |
| Description | 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 Calderon type integral operator patterned on the Gaussian isoperimetric function. Consequently, the best possible norms in a variety of spe- cific 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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