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It is well-known that the embedding of the Sobolev space of weaklydifferentiable functions into H\"{o}lder spaces holds if the integrabilityexponent is higher than the space dimension. In this paper, the embedding ofthe Sobolev functions into the H\"{o}lder spaces is expressed in terms of theminimal weak differentiability requirement independent of the integrabilityexponent. The proof is based on the generalization of the Newton-Leibnizformula to the $n$-dimensional rectangle and inductive application of theSobolev trace embedding results. The method is applied to prove the embeddingof the Sobolev spaces with dominating mixed smoothness into H\"{o}lder spaces.

Generalized Newton-Leibniz Formula and the Embedding of the Sobolev Functions with Dominating Mixed Smoothness into H\"{o}lder Spaces