Regularity theory for the fractional harmonic oscillator

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.Comment: 23 pages, references added, to appear in Journal of Functional Analysi

On the boundary convergence of solutions to the Hermite-Schr\"odinger equation

In the half-space $\mathbb{R}^d \times \mathbb{R}_+$, we consider the Hermite-Schr\"odinger equation $i\partial u/\partial t = - \Delta u + |x|^2 u$, with given boundary values on $\mathbb{R}^d$. We prove a formula that links the solution of this problem to that of the classical Schr\"odinger equation. It shows that mixed norm estimates for the Hermite-Schr\"odinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary, by means of this link.Comment: 12 page