Regularity theory for the fractional harmonic oscillator

Abstract

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 $C_H^{k,\alpha }$ 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 Hölder spaces $C_H^{k,\alpha }$ 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.

23 pages, references added, to appear in Journal of Functional Analysis

References in corpus (1)

• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian