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Harmonic measures for symmetric stable processes

Jang-Mei Wu (2002)

Studia Mathematica

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 S that determine whether ω(S,D) is zero or positive.

Harmonic spaces associated with adjoints of linear elliptic operators

Peter Sjögren (1975)

Annales de l'institut Fourier

Let L be an elliptic linear operator in a domain in 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.

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