Explore dimension-free Sobolev-type embeddings in the Gauss space in this 51-minute lecture by Lenka Slavíková from the Hausdorff Center for Mathematics. Delve into the logarithmic Sobolev inequality of Gross and its implications for functions on ℝ^n with first-order weak derivatives in L^2(ℝ^n,γ_n). Examine extensions of Gross' inequality, focusing on Orlicz spaces and higher-order derivatives. Investigate the relationship between these results and the Gaussian isoperimetric inequality. Compare Gaussian Sobolev embeddings to their Euclidean counterparts on ℝ^n with standard Lebesgue measure. Gain insights into advanced mathematical concepts and their applications in functional analysis and probability theory.