Gradient estimates of q-harmonic functions of fractional Schrodinger operator

TL;DR

This paper establishes gradient estimates for q-harmonic functions related to the fractional Schrödinger operator with fractional order α in (0,1], revealing conditions under which the gradient exists and providing bounds, using probabilistic methods.

Contribution

It provides new gradient estimates for fractional Schrödinger operators with α in (0,1], extending known results and identifying the critical Hölder continuity threshold for the potential q.

Findings

Gradient of nonnegative solutions exists under Hölder continuity of q with order > 1 - α.

Gradient estimates are valid for α in (0,1], using probabilistic techniques.

Weak solutions are shown to be strong solutions for α in (0,1).

Abstract

We study gradient estimates of $q$-harmonic functions $u$ of the fractional Schr{\"o}dinger operator $\Delta^{\alpha/2} + q$, $\alpha \in (0,1]$ in bounded domains $D \subset \R^d$. For nonnegative $u$ we show that if $q$ is H{\"o}lder continuous of order $\eta > 1 - \alpha$ then $\nabla u(x)$ exists for any $x \in D$ and $|\nabla u(x)| \le c u(x)/ (\dist(x,\partial D) \wedge 1)$. The exponent $1 - \alpha$ is critical i.e. when $q$ is only $1 - \alpha$ H{\"o}lder continuous $\nabla u(x)$ may not exist. The above gradient estimates are well known for $\alpha \in (1,2]$ under the assumption that $q$ belongs to the Kato class $\calJ^{\alpha - 1}$. The case $\alpha \in (0,1]$ is different. To obtain results for $\alpha \in (0,1]$ we use probabilistic methods. As a corollary, we obtain for $\alpha \in (0,1)$ that a weak solution of $\Delta^{\alpha/2}u + q u = 0$ is in fact a strong solution.