Обобщение одной теоремы И. И. Привалова

Записки научных семинаров ПОМИ РАН. 2025. № 545. С. 145–156.

Колпаков А. С., Shirokov N. A.

Let E ⊂C be a compact set. The set E is called an Ahlfors–David set of dimension θ, 0 < θ < 2, if there exist constants C1, C2 > 0 such that for any z ∈ E and arbitrary r, 0 < r < diamE, one has C1rθ Λθ E∩Br(z) C2rθ, where Λθ(S) means the Hausdorff measure of dimension θ and Br(z) def = {ζ : |ζ −z| r}. Let Γ be a closed Jordan curve that is an Ahlfors–David set of dimen sion 1 +α, 0 < α < 1, which bounds a domain D; we put G = C\D. We introduce the spaces Hβ D and Hβ G as the spaces of functions g or h analytic respectively in D or G, h(∞) = 0, and satisfying the β-H¨older condition in D or G. We denote by Hβ(Γ) the space of all complex-valued functions f defined on Γ and satisfying the β-H¨older condition on it. We prove the following result. Theorem.Assume that Γ is a Jordan curve as described above, α < β < 1, f ∈Hβ(Γ). Then there exist functions g ∈ Hβ D and h ∈ Hβ G such that f(z) = g(z)+h(z), z ∈ Γ

Keywords: analytic functions, Hölder classes, аналитические функции, the Ahlfors–David condition, классы Гёльдера, условие Альфорса–Давида