Regularity theory for the fractional harmonic oscillator

In this paper we develop the theory of Schauder estimates for the fractional harmonic oscillator Hσ=σ(−Δ+2|x|), 0<σ<1. More precisely, a new class of smooth functions is defined, in which we study the action of Hσ. In fact these spaces are those adapted to the operator H, hence 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 Hölder spaces is needed, that we believe of independent interest.