On the H\"older regularity for the fractional Schr\"odinger equation and its improvement for radial data

TL;DR

This paper investigates the local H"older regularity of solutions to the fractional Schr"odinger equation, showing an improved regularity for radial data that effectively reduces the problem to a one-dimensional setting.

Contribution

It establishes a regularity improvement for solutions with radial data, reducing the H"older exponent from a multi-dimensional to a one-dimensional form.

Findings

H"older exponent for solutions is 2s - N/p in general.

For radial data, the exponent improves to 2s - 1/p away from the origin.

Similar regularity improvements apply to the gradient of solutions.

Abstract

We consider the linear, time-independent fractional Schr\"odinger equation $$ (-\Delta)^s \psi+V\psi=f. $$ We are interested in the local H\"older 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 H\"older 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 H\"older exponent improves from $2s-N/p$ to $2s-1/p$ away from the origin. Similar results hold for $\nabla\psi$ as well.