Optimal Gaussian Sobolev embeddings |
| |
Authors: | Andrea Cianchi Luboš Pick |
| |
Institution: | a Dipartimento di Matematica e Applicazioni per l'Architettura, Università di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy b Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic |
| |
Abstract: | A reduction theorem is established, showing that any Sobolev inequality, involving arbitrary rearrangement-invariant norms with respect to the Gauss measure in Rn, is equivalent to a one-dimensional inequality, for a 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. |
| |
Keywords: | Logarithmic Sobolev inequalities Gauss measure Sobolev embeddings Rearrangement-invariant spaces Optimal domain Optimal range Orlicz spaces Lorentz spaces Hardy operators involving suprema |
本文献已被 ScienceDirect 等数据库收录! |
|