On the boundary behavior of positive solutions of elliptic differential equations

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)t^α → A as t → +0 if and only if μ(B_r(x))/r^n-1 r^α → C_αA as r → +0, where C_α = π^n/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.