Embeddings of anisotropic Sobolev spaces into spaces of anisotropic H\"{o}lder-continuous functions

TL;DR

This paper develops a new framework for embedding anisotropic variable exponent Sobolev spaces into anisotropic variable exponent Hölder-continuous function spaces, enhancing understanding of function regularity in anisotropic settings.

Contribution

It introduces a foundational approach to extend Hölder continuity to anisotropic variable exponent spaces, broadening the theoretical understanding of these function spaces.

Findings

Established embedding theorems for anisotropic variable exponent Sobolev and Hölder spaces

Provided new insights into the regularity properties of anisotropic functions

Opened pathways for applications in mathematical and applied sciences

Abstract

We introduce a novel framework for embedding anisotropic variable exponent Sobolev spaces into spaces of anisotropic variable exponent H\"{o}lder-continuous functions within rectangular domains. We establish a foundational approach to extend the concept of H\"{o}lder continuity to anisotropic settings with variable exponents, providing deeper insight into the regularity of functions across different directions. Our results not only broaden the understanding of anisotropic function spaces but also open new avenues for applications in mathematical and applied sciences.