Symmetrization, symmetric stable processes, and Riesz capacities

Let be a symmetric -stable process killed on exiting an open subset of . We prove a theorem that describes the behavior of its transition probabilities under polarization. We show that this result implies that the probability of hitting a given set in the complement of in the first exit moment from increases when and are polarized. It can also lead to symmetrization theorems for hitting probabilities, Green functions, and Riesz capacities. One such theorem is the following: Among all compact sets in with given volume, the balls have the least -capacity ( ).