Harmonic measures for symmetric stable processes

Let D be an open set in ℝⁿ (n ≥ 2) and ω(., D) be the harmonic measure on D^c with respect to the symmetric α-stable process (0 < α < 2) killed upon leaving D. We study inequalities on volumes or capacities which imply that a set 5 on ∂D has zero harmonic measure and others which imply that S has positive harmonic measure. In general, it is the relative sizes of the sets S and D^c \ S that determine whether ω(S, D) is zero or positive.