Some remarks on Campanato Theorem and the Anisotropic Bessel Spaces

TL;DR

This paper extends the Campanato Theorem to anisotropic settings, demonstrating that anisotropic Bessel spaces embed into Holder spaces, and constructs fundamental solutions for anisotropic fractional Laplacians.

Contribution

It introduces an anisotropic version of the Campanato Theorem and establishes embeddings of anisotropic Bessel spaces into Holder spaces, with applications to fractional Laplacian operators.

Findings

Anisotropic Campanato Theorem established

Continuous embedding of anisotropic Bessel spaces into Holder spaces

Fundamental solutions for anisotropic fractional Laplacians constructed

Abstract

In this paper, we establish an anisotropic version of Campanato Theorem and show that the anisotropic Bessel spaces are continuously embedded in the spaces of Holder continuous functions. As an application of this embedding, we build fundamental solutions for a class of anisotropic fractional Laplacian operators.