Gradient estimates of q-harmonic functions of fractional Schrodinger operator

Abstract

We study gradient estimates of $q$-harmonic functions $u$ of the fractional Schr{ö}dinger operator ${\Delta }^{\alpha /2}+q$, $\alpha \in (0,1]$ in bounded domains $D\subset {\mathbb{R}}^d$. For nonnegative $u$ we show that if $q$ is H{ö}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{ö}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 $\calJ^{α- 1}$. 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 ${\Delta }^{\alpha /2}u+qu=0$ is in fact a strong solution.

References in corpus (1)

• Geometric property of the Ground State Eigenfunction for Cauchy Process