# Extension and embedding theorems for Campanato spaces on $C^{0,\gamma}$ domains

## TL;DR

This paper studies Campanato spaces on $C^{0,eta}$ domains, establishing embedding theorems and extension results that preserve the space parameters, thus advancing understanding of function regularity in anisotropic settings.

## Contribution

It introduces extension and embedding theorems for Campanato spaces on $C^{0,eta}$ domains with anisotropic metrics, preserving exponents during extension.

## Key findings

- Established Campanato embedding theorem for $C^{0,eta}$ domains.
- Proved functions in these spaces can be extended to Euclidean space without losing properties.
- Analyzed anisotropic metric effects on function space embeddings.

## Abstract

We consider Campanato spaces with exponents $\lambda , p$ on domains of class $C^{0,\gamma}$ in the N-dimensional Euclidean space endowed with a natural anisotropic metric depending on $\gamma$. We discuss several results including the appropriate Campanato's embedding theorem and we prove that functions of those spaces can be extended to the whole of the Euclidean space without deterioration of the exponents $\lambda, p$.