Optimal rearrangement invariant Sobolev embeddings in mixed norm spaces

TL;DR

This paper enhances Sobolev embeddings in rearrangement invariant spaces by identifying optimal domains and ranges, and demonstrates that mixed norm spaces can strengthen classical Sobolev estimates.

Contribution

It introduces improved Sobolev-type embeddings in rearrangement invariant spaces with optimal domain and range characterization, extending classical results with mixed norm spaces.

Findings

Classical Sobolev estimates can be strengthened using mixed norm target spaces.

Optimal embeddings between rearrangement invariant and mixed norm spaces are established.

The results generalize and improve known Sobolev embedding theorems.

Abstract

We improve the Sobolev-type embeddings due to Gagliardo and Nirenberg in the setting of rearrangement invariant (r.i.) spaces. In particular we concentrate on seeking the optimal domains and the optimal ranges for these embeddings between r.i. spaces and mixed norm spaces. As a consequence, we prove that the classical estimate for the standard Sobolev space by Poornima, O'Neil and Peetre (1 <=p<n), and by Hansson, Brezis and Wainger and Maz'ya (p=n) can be further strengthened by considering mixed norms on the target spaces.