Let u be a positive harmonic function in the unit ball B1 ⊂ Rn, and let μ be the boundary measure of u. For a point x ∈ ∂B1, let n(x) denote the unit inner normal at x. Let α be a number in (-1, n - 1], and let A ∈ [0,+∞). In the paper, it is proved that u(x+n(x)t)tα → A as t → +0 if and only if μ(Br(x))/rn-1 rα → CαA as r → +0, where Cα = πn/2/Γ(n-α+1/2)Γ(α+1/2). For α = 0, this follows from the theorems by Rudin and Loomis that claim that a positive harmonic function has a limit along the normal if and only if the boundary measure has the derivative at the corresponding point of the boundary. For α = n-1, this is related to the size of the point mass of μ at x and in this case the claim follows from the Beurling minimum principle. For the general case of α ∈ (-1, n - 1), the proof employs the Wiener Tauberian theorem in a way similar to Rudin's approach. In dimension 2, conformal mappings can be used to generalize the statement to sufficiently smooth domains; in dimension n ≥ 3 it is shown that this generalization is possible for α ∈ [0, n-1] due to harmonic measure estimates. A similar method leads to an extension of results by Loomis, Ramey, and Ullrich on nontangential limits of harmonic functions to positive solutions of elliptic differential equations with Hölder continuous coefficients.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Applied Mathematics
- Harmonic functions
- Tauberian theorems