Abstract:
Let $E\subset \mathbb{C}$ be a compact set. The set $E$ is called an Ahlfors–David set of dimension $\theta$, $0<\theta<2$, if there exist constants $C_1$, $C_2>0$ such that for any $z\in E$ and arbitrary $r$, $0<r<\mathrm{diam}\, E$, one has $$ C_1r^{\theta}\leqslant\Lambda_{\theta}(E\cap\overline{B}_r(z))\leqslant C_2r^{\theta}, $$ where $\Lambda_{\theta}(S)$ means the Hausdorff measure of dimension $\theta$ and $\overline{B}_r(z)\overset{\rm def}{=}\{\zeta: \left|\zeta-z\right|\leqslant r\}$.
Let $\Gamma$ be a closed Jordan curve that is an Ahlfors–David set of dimension $1+\alpha,$$0<\alpha<1,$ which bounds a domain $D$; we put $G=\mathbb{C}\backslash \overline{D}$. We introduce the spaces $H^{\beta}(\overline{D})$ and $H^{\beta}(\overline{G})$ as the spaces of functions $g$ or $h$ analytic respectively in $D$ or $G$, $h(\infty)=0$, and satisfying the $\beta$-Hölder condition in $\overline{D}$ or $\overline{G}$. We denote by $H^{\beta}(\Gamma)$ the space of all complex-valued functions $f$ defined on $\Gamma$ and satisfying the $\beta$-Hölder condition on it. We prove the following result.
Theorem.Assume that$\Gamma$is a Jordan curve as described above, $\alpha<\beta<1,$$f\in H^{\beta}(\Gamma).$Then there exist functions$g\in H^{\beta}(\overline{D})$and$h \in H^{\beta}(\overline{G})$such that $$ f(z)=g(z)+h(z),\ z\in\Gamma. $$
Key words and phrases:analytic functions, Hölder classes, Ahlfors–David sets.
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