Optimal Sobolev embeddings for the Ornstein-Uhlenbeck operator

A comprehensive analysis of Sobolev-type inequalities for the Ornstein-Uhlenbeck operator in Gauss space is offered. unified approach proposed, providing one with criteria their validity class rearrangement-invariant function norms. Optimal target and domain norms relevant are characterized via a reduction principle to one-dimensional Calderón type integral patterned on Gaussian isoperimetric function. Consequently, best possible variety specific families spaces, including Lebesgue, Lorentz, Lorentz-Zygmund, Orlicz Marcinkiewicz detected. The hinges preliminary discussion existence uniqueness generalized solutions equations, space, operator, just integrable right-hand side. decisive role also played by pointwise estimate, rearrangement form, these solutions.