On some refinements of the embedding of critical Sobolev spaces into BMO

TL;DR

This paper introduces non-homogeneous analogs of Van Schaftingen's classes, refining the embedding of critical Sobolev spaces into BMO, with results applicable on bounded Lipschitz domains and Riemannian manifolds.

Contribution

It develops new non-homogeneous classes that improve the understanding of Sobolev space embeddings into BMO, extending previous results to broader geometric settings.

Findings

Refined embedding of $W^{1,n}$ into BMO using new classes

Extension of results to Lipschitz domains and Riemannian manifolds

Establishment of non-homogeneous analogs of Van Schaftingen's classes

Abstract

We introduce the non-homogeneous analogs of Van Schaftingen's classes. We show that these classes refine the embedding $W^{1,n}\subset bmo$. The analogous results established on bounded Lipschitz domains and Riemannian manifolds with bounded geometry.