Regularity theory for the fractional harmonic oscillator

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 $C^{k,α}_H$ is defined, in which we study the action of $H^σ$. 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 Hölder spaces $C^{k,α}_H$ is needed, that we believe of independent interest. The parallel results for the fractional powers of the Laplacian $(-Δ)^σ$ were applied by Caffarelli, Salsa and Silvestre to the study of the regularity of the obstacle problem for the fractional Laplacian.

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