  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.
