Optimal function spaces and Sobolev embeddings
Functional Analysis
2024-07-10 v1
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.
Cite
@article{arxiv.2407.06307,
title = {Optimal function spaces and Sobolev embeddings},
author = {David Kubíček},
journal= {arXiv preprint arXiv:2407.06307},
year = {2024}
}