  In this paper we develop the theory of Schauder estimates for the fractional
harmonic oscillator $H^\sigma=(-\Delta+|x|^2)^\sigma$, $0<\sigma<1$. More
precisely, a new class of smooth functions $C^{k,\alpha}_H$ is defined, in
which we study the action of $H^\sigma$. It turns out that these spaces are the
suited ones for this type of regularity estimates. In order to prove our
results, an analysis of the interaction of the Hermite-Riesz transforms with
the H\"older spaces $C^{k,\alpha}_H$ is needed, that we believe of independent
interest.
  The parallel results for the fractional powers of the Laplacian
$(-\Delta)^\sigma$ were applied by Caffarelli, Salsa and Silvestre to the study
of the regularity of the obstacle problem for the fractional Laplacian.
