TY - JOUR

T1 - Ahlfors-Type Theorem for Hausdorff Measures

AU - Florinskiy, A.A.

AU - Fofanov, K.A.

AU - Shirokov, N.A.

N1 - Export Date: 19 October 2024 Адрес для корреспонденции: Fofanov, K.A.; Herzen State Pedagogical UniversityRussian Federation; эл. почта: kirfof@mail.ru

PY - 2024/9/27

Y1 - 2024/9/27

N2 - Suppose that Δ ⊂ C is a domain, f is an analytic function in Δ, D = f(Δ) is considered as a Riemann surface. Put lR = {z ∈ Δ : |f(z)| = R}. Let E ⊂ Δ be a closed set. Put hα,β(r) = rα| 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, ∂Δ) ≥ ε, |z| ≤ 1ε} and TR,ε = f(lR,ε ∩ E), TR,ε ⊂ D. Put (Formula presented.) Define the upper Lebesgue integral ∫∞∗0g 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 put ∫∞∗0g dm =def∫∞0Hydy. 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 (Formula presented.) © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.

AB - Suppose that Δ ⊂ C is a domain, f is an analytic function in Δ, D = f(Δ) is considered as a Riemann surface. Put lR = {z ∈ Δ : |f(z)| = R}. Let E ⊂ Δ be a closed set. Put hα,β(r) = rα| 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, ∂Δ) ≥ ε, |z| ≤ 1ε} and TR,ε = f(lR,ε ∩ E), TR,ε ⊂ D. Put (Formula presented.) Define the upper Lebesgue integral ∫∞∗0g 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 put ∫∞∗0g dm =def∫∞0Hydy. 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 (Formula presented.) © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.

UR - https://www.mendeley.com/catalogue/a561910a-b30e-3969-9fae-ed9f220ec490/

U2 - 10.1007/s10958-024-07395-4

DO - 10.1007/s10958-024-07395-4

M3 - статья

VL - 284

SP - 880

EP - 893

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 6

ER -