The fractional Bessel equation in Hölder spaces

Motivated by the Poisson equation for the fractional Laplacian on the whole space with radial right hand side, we study global Hölder 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 Hölder 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.