Optimal function spaces and Sobolev embeddings

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.