Fatou's theorem for subordinate Brownian motions with Gaussian components on $C^{1,1}$ open sets

TL;DR

This paper establishes Fatou's theorem for nonnegative harmonic functions related to subordinate Brownian motions with Gaussian components on smooth bounded domains, showing nontangential convergence almost everywhere and mutual absolute continuity of harmonic and surface measures.

Contribution

It extends Fatou's theorem to a class of subordinate Brownian motions with Gaussian components on $C^{1,1}$ domains, including boundary measure properties.

Findings

Nontangential convergence holds almost everywhere.

Harmonic measure is mutually absolutely continuous with surface measure.

Tangential convergence fails on the unit ball.

Abstract

We prove Fatou's theorem for nonnegative harmonic functions with respect to subordinate Brownian motions with Gaussian components on bounded $C^{1,1}$ open sets $D$. We prove that nonnegative harmonic functions with respect to such processes on $D$ converge nontangentially almost everywhere with respect to the surface measure as well as the harmonic measure restricted to the boundary of the domain. In order to prove this, we first prove that the harmonic measure restricted to $\partial D$ is mutually absolutely continuous with respect to the surface measure. We also show that tangential convergence fails on the unit ball.