The fractional Bessel equation in H\"older spaces

TL;DR

This paper develops regularity estimates for the fractional Bessel equation in H"older spaces, using semigroup methods and Bessel harmonic extensions, with applications to radial solutions of the fractional Laplacian.

Contribution

It introduces a novel approach employing semigroup language and Campanato-type spaces to derive regularity estimates for fractional Bessel equations.

Findings

Established pointwise formulas for fractional operators via Bessel heat solutions.

Characterized H"older spaces through Bessel harmonic extensions and fractional Carleson measures.

Derived regularity estimates for radial solutions to the fractional Laplacian.

Abstract

Motivated by the Poisson equation for the fractional Laplacian on the whole space with radial right hand side, we study global H\"older and Schauder estimates for a fractional Bessel equation. Our methods stand on the so-called semigroup language. Indeed, by using the solution to the Bessel heat equation we derive pointwise formulas for the fractional operators. Appropriate H\"older spaces, which can be seen as Campanato-type spaces, are characterized through Bessel harmonic extensions and fractional Carleson measures. From here the regularity estimates for the fractional Bessel equations follow. In particular, we obtain regularity estimates for radial solutions to the fractional Laplacian.