Generalized Newton-Leibniz Formula and the Embedding of the Sobolev Functions with Dominating Mixed Smoothness into Hölder Spaces

Abstract: It is well-known that the embedding of the Sobolev space of weakly differentiable functions into H\"{o}lder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions into the H\"{o}lder spaces is expressed in terms of the minimal weak differentiability requirement independent of the integrability exponent. The proof is based on the generalization of the Newton-Leibniz formula to the $n$-dimensional rectangle and inductive application of the Sobolev trace embedding results. The method is applied to prove the embedding of the Sobolev spaces with dominating mixed smoothness into H\"{o}lder spaces.