Intrinsic Ultracontractivity and Conditional Gauge for Symmetric Stable Processes

It is shown in this paper that the conditional gauge theorem holds for symmetricα-stable processes on boundedC1, 1domains inRnwhere 0<α<2 andn⩾2. Two of the major tools used to prove this conditional gauge theorem are logarithmic Sobolev inequality and intrinsic ultracontractivity.