Теорема типа Л. Альфорса для мер Хаусдорфа

Записки научных семинаров ПОМИ РАН. 2023. Т. 527. С. 221–241.

Флоринский А. А., Фофанов К. А., Shirokov N. A.

Suppose that ∆ ⊂ C is a domain, a function f is analytic in ∆, D = f(∆) is viewed as a Riemann surface. We put lR = {z ∈ ∆ : |f(z)| = R}. Let E ⊂ ∆ be a closed set. Put hα,β(r) = r α| ln r| β , 0 < α < 1, 0 < β < 1. Let Λα,β(·), Λα+1,β(·) be the Hausdorff measures with respect to the functions hα,β, hα+1,β. Assume that Λα+1,β(E) < ∞. We introduce the sets lR,ε = {z ∈ lR : dist(z, ∂∆) > ε, |z| 6 1 ε } and TR,ε = f(lR,ε ∩ E), TR,ε ⊂ D. Put Gε(R) =    0 if Λα,β(TR,ε) = 0 or Λα,β(TR,ε) = ∞, Λ 1+α α α,β (E∩lR,ε) Λ 1 α α,β(TR,ε) if 0 < Λα,β(TR,ε) < ∞.

We define the upper Lebesgue integral R∞∗ 0 g dm for a function g, g(x)>0, x > 0 in the following way: let U(y) def = {x > 0 : g(x) > y}, H(y) = m∗U(y). Then we put R∞∗ 0 g dm def = R∞ 0 H(y)dy. We prove the following result. Theorem. The condition Λα,β(TR,ε) < ∞ is fulfilled for almost all R with respect to the 1-Lebesgue measure and Z∞∗ 0 lim ε→+0 Gε(R)dR 6 2Λ1+α,β(E).

Keywords: Riemann surfaces, Римановы поверхности, length-area principle, Hausdorff measures, принцип длины и площади, меры Хаусдорфа