## Illinois Journal of Mathematics

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

Hyunchul Park

#### Abstract

We prove Fatou’s theorem for nonnegative harmonic functions with respect to killed 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.

#### Article information

Source
Illinois J. Math., Volume 60, Number 3-4 (2016), 761-790.

Dates
Revised: 3 April 2017
First available in Project Euclid: 22 September 2017

https://projecteuclid.org/euclid.ijm/1506067290

Digital Object Identifier
doi:10.1215/ijm/1506067290

Mathematical Reviews number (MathSciNet)
MR3705444

Zentralblatt MATH identifier
1376.31003

#### Citation

Park, Hyunchul. Fatou’s theorem for subordinate Brownian motions with Gaussian components on $C^{1,1}$ open sets. Illinois J. Math. 60 (2016), no. 3-4, 761--790. doi:10.1215/ijm/1506067290. https://projecteuclid.org/euclid.ijm/1506067290

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