Regularity theory for the fractional harmonic oscillator

TL;DR

This paper develops Schauder estimates for the fractional harmonic oscillator, introducing new smooth function spaces suited for regularity analysis and exploring the interaction of Hermite-Riesz transforms with these spaces.

Contribution

It introduces a novel class of smooth functions $C^{k,eta}_H$ for analyzing fractional harmonic oscillators, expanding regularity theory in this context.

Findings

Defined new smooth function spaces $C^{k,eta}_H$ for fractional harmonic oscillator analysis

Analyzed Hermite-Riesz transforms' interaction with these spaces

Established Schauder estimates for $H^\sigma$ in the new function spaces

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 $C^{k,\alpha}_H$ 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.