Sharp Sobolev type embeddings on the entire Euclidean space

2017/11/30 by Alberico, Angela, Cianchi, Andrea, Pick, Lubos +1 · 1 citation
#46E30 #46E35 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1711.11549

Abstract

A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement- invariant norms on the entire Euclidean space Rⁿ, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.

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• Uniform decay of function norms

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