Optimal Gaussian Sobolev embeddings

A reduction theorem is established, showing that my Sobolev inequality, involving arbitrary rearrangement-invariant norms with respect to the Gauss measure in R-n, is equivalent to a one-dimensional inequality, for it suitable Hardy-type operator, involving the same norms with respect to the standard Lebesgue measure on the unit interval. This result is exploited to provide a general characterization of optimal range and domain norms in Gaussian Sobolev inequalities. Applications to special instances yield optimal Gaussian Sobolev inequalities in Orlicz and Lorentz(-Zygmund) spaces, point out new phenomena, such as the existence of self-optimal spaces, and provide further insight into classical results. (C) 2009 Elsevier Inc. All rights reserved.