The celebrated logarithmic Sobolev inequality of Gross asserts that if u is a function on ℝ^n whose first-order weak derivatives belong to the space L^2(ℝ^n,γ_n), where dγ_n(x)=(2𝛑)^{-n/2} e^{-|x|^2/2} dx stands for the Gauss measure on ℝ^n, then |u|^2 log_+ |u| is integrable with respect to γ_n, i.e., u belongs to the Orlicz space L^2 log L(ℝ^n,γ_n). In addition, the corresponding estimate of the L^2 log L-norm of u by its L^2-based Sobolev norm holds with a constant independent of the dimension n.
In this talk, I will discuss extensions of Gross' inequality in which L^2(ℝ^n,γ_n) is replaced by a more general function space, with a particular emphasis on the class of Orlicz spaces. Variants of these results involving higher-order derivatives will be considered as well. I will explain how the presented results relate to the Gaussian isoperimetric inequality. I will also compare the Gaussian Sobolev embeddings to their counterparts on the Euclidean space ℝ^n equipped with the standard Lebesgue measure.
Негізгі бет Lenka Slavíková: Dimension-free Sobolev-type embeddings in the Gauss space
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