Sharp Sobolev type embeddings on the entire Euclidean space

TL;DR

This paper establishes sharp Sobolev-type embeddings on the entire Euclidean space using rearrangement-invariant norms, introduces a reduction principle to simplify the analysis, and applies results to Orlicz-Sobolev and Lorentz-Sobolev spaces.

Contribution

It provides the first sharp Sobolev embeddings on the whole Euclidean space for general rearrangement-invariant norms, with a new reduction principle simplifying the problem.

Findings

Identified the optimal target space for embeddings on R^n.

Developed a reduction principle linking embeddings to one-dimensional inequalities.

Applied results to Orlicz-Sobolev and Lorentz-Sobolev spaces.

Abstract

A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement- invariant norms on the entire Euclidean space R^n, 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.