On the Hölder regularity for the fractional Schrödinger equation and its improvement for radial data

Abstract

We consider the linear, time-independent fractional Schrödinger equation $$(-\Delta {)}^s\psi +V\psi =f.$$ We are interested in the local Hölder exponents of distributional solutions $\psi$, assuming local $L^p$ integrability of the functions $V$ and $f$. By standard arguments, we obtain the formula $2s-N/p$ for the local Hölder exponent of $\psi$ where we take some extra care regarding endpoint cases. For our main result, we assume that $V$ and $f$ (but not necessarily $\psi$) are radial functions, a situation which is commonplace in applications. We find that the regularity theory "becomes one-dimensional" in the sense that the Hölder exponent improves from $2s-N/p$ to $2s-1/p$ away from the origin. Similar results hold for $\nabla \psi$ as well.

30 pages