Superharmonicity of the fractional ground state

Abstract

We prove that the positive ground state $u$ of the fractional Laplacian $(-\Delta {)}^s$, $s\in (0,1)$, on an arbitrary open set $\Omega$ satisfies $-\Delta u>{\lambda }_s(\Omega {)}^{1/s}u$ in $\Omega$. In one dimension, this yields strong concavity and settles a conjecture of Bañuelos, Kulczycki, and Méndez-Hernández. We establish a hierarchy of pointwise inequalities comparing different powers of the Laplacian. In balls, these inequalities give a quantitative Hessian estimate for $\log u$, and hence log-concavity in every dimension. By contrast, we show that log-concavity fails in general convex domains by constructing counterexamples in sufficiently thin ellipsoids. We also extend the superharmonicity principle to ground states of $\psi (-\Delta )$ for complete Bernstein functions $\psi$.