Optimal function spaces and Sobolev embeddings

TL;DR

This paper links supremum operator boundedness to the optimality of Sobolev embedding function spaces, providing explicit computation methods and applying results to specific domain classes and product spaces.

Contribution

It introduces a new approach connecting supremum operators with Sobolev space optimality, and offers explicit formulas for optimal norms in embeddings.

Findings

Explicit computation of optimal domain and target norms in Sobolev embeddings

Application of results to John domains and Maz'ya class domains

Partial applicability to embeddings in product probability spaces

Abstract

We establish equivalence between the boundedness of specific supremum operators and the optimality of function spaces in Sobolev embeddings acting on domains in ambient Euclidean space with a prescribed isoperimetric behavior. Our approach is based on exploiting known relations between higher-order Sobolev embeddings and isoperimetric inequalities. We provide an explicit way to compute both the optimal domain norm and the optimal target norm in a Sobolev embedding. Finally, we apply our results to higher-order Sobolev embeddings on John domains and on domains from the Maz'ya classes. Furthermore, our results are partially applicable to embeddings involving product probability spaces.