Regularity theory for the fractional harmonic oscillator

摘要

In this paper we develop the theory of Schauder estimates for the fractional harmonic oscillator H~σ=(-Δ+|x|~2)~σ, 0<σ<1. More precisely, a new class of smooth functions CH~(k,α) is defined, in which we study the action of Hα. 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 H?lder spaces C_H~(k,α) is needed, that we believe of independent interest.