Inverse Theorem for Approximation on Subsets of a Domain with Cusps

Let \(\mathfrak{P}\left(z\right)\) $\mathfrak{P}\left(z\right)$ be a doubly periodic Weierstrass function with periods 2ω₁, 2ω₂, and let Q be the parallelogram of periods, Q = {z ∈ \({\mathbb{C}}\) $\mathbb{C}$ : z = 2α₁ω₁+2α₂ω₂, α₁, α₂ ∈ [0, 1)}. We consider a simply connected domain D, \(\overline{D }\) $\overline{D}$ ⊂ Q, such that its boundary ∂D contains cusps, and a function f that is analytic in D and continuous on ∂D. We assume that the modulus of continuity ω(t) satisfies the relation \(\underset{0}{\overset{x}{\int }}\frac{\omega \left(t\right)}{t}dt+x\underset{x}{\overset{\infty }{\int }}\frac{\omega \left(t\right)}{{t}^{2}}dt\le c\omega \left(x\right).\) $$\underset{0}{\stackrel{x}{\int }}\frac{\omega \left(t\right)}{t}dt+x\underset{x}{\stackrel{\infty }{\int }}\frac{\omega \left(t\right)}{t^2}dt\le c\omega \left(x\right).$$

Let Φ map conformally the domain \({\mathbb{C}}\backslash D\) $\mathbb{C}\backslash D$ onto \({\mathbb{C}}\backslash {\mathbb{D}}\) $\mathbb{C}\backslash \mathbb{D}$ with the normalization Φ(∞) = ∞,Φ′(∞) > 0. We put L_1+t = {z ∈ \({\mathbb{C}}\backslash D\) $\mathbb{C}\backslash D$ : |Φ(z)| = 1+t}, δ_n(z) = dist(z, \({L}_{1+ \frac{1}{n}}\) $L_{1+\frac{1}{n}}$ ), z ∈ ∂D. The main result of the paper is the following statement.

Theorem 1. Assume that there exists a sequence of polynomials P_n(u, v), deg P_n ≤ n, such that \(\begin{array}{cc}\left|f\left(z\right)-{P}_{n}\left(\mathfrak{P}\left(z\right),{\mathfrak{P}}^{\prime}\left(z\right)\right)\right|\le {C\delta }_{n}^{r}\left(z\right)\omega \left({\delta }_{n}\left(z\right)\right),& z\in \partial D\end{array},\) $$\begin{array}{cc}\left|f,\left(z\right),-,P_n,\left(\mathfrak{P},\left(z\right),,,{\mathfrak{P}}^{\prime },\left(z\right)\right)\right|\le {C\delta }_n^r\left(z\right)\omega \left({\delta }_n,\left(z\right)\right),& z\in \partial D\end{array},$$

C is independent of n and z. Then f ∈ H^r+ω(D).