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

TL;DR

This paper generalizes the Newton-Leibniz formula to higher dimensions to characterize the embedding of Sobolev functions with mixed smoothness into Hölder spaces, independent of integrability exponents.

Contribution

It introduces a new generalized Newton-Leibniz formula and applies it to establish embeddings of Sobolev spaces with dominating mixed smoothness into Hölder spaces.

Findings

Embedding characterized by minimal weak differentiability

Generalized Newton-Leibniz formula for n-dimensional rectangles

Proof of embedding for Sobolev spaces with mixed smoothness

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.