Fatou's Theorem for censored stable processes

We give a proof of Fatou's Theorem for censored [alpha]-stable processes in a bounded C1,1 open set D where [alpha][set membership, variant](1,2). As an application of Fatou's Theorem, we show that the harmonic measure for such censored [alpha]-stable process is mutually absolutely continuous with respect to the surface measure of [not partial differential]D. Fatou's Theorem is also established for operators obtained from the generator of the censored [alpha]-stable process through non-local Feynman-Kac transforms. Fatou's Theorem for censored relativistic stable processes is also true as a consequence.