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 S 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 $D^c\setminus S$ that determine whether ω(S,D) is zero or positive.

Let $L$ be an elliptic linear operator in a domain in ${\mathbf{R}}^n$. We imposse only weak regularity conditions on the coefficients. Then the adjoint $L^{*}$ exists in the sense of distributions, and we start by deducing a regularity theorem for distribution solutions of equations of type $L^{*}u=$ given distribution. We then apply to $L^{*}$ R.M. Hervé’s theory of adjoint harmonic spaces. Some other properties of $L^{*}$ are also studied. The results generalize earlier work of the author.