Inverse theorem for approximation on subsets of a domain with cups

K. А. Sintsova

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 51, Tome 527 (2023), pp. 204-220

Résumé

Let $\mathfrak{P}(z)$ be a doubly periodic Weierstrass function with periods $2\boldsymbol{\omega}_1$, $2\boldsymbol{\omega}_2$, and let $ Q$ be the parallelogramm of periods, $Q = \{z \in \mathbb{C}$ $: z = 2\alpha_1\boldsymbol{\omega}_1 + 2\alpha_2\boldsymbol{\omega}_2, \alpha_1, \alpha_2 \in [0,1)\}$. We consider a simply connected domain $D, \overline{D} \subset Q$, such that its boundary $\partial D$ contains cusps, and a function $f$ that is analytic in $D$ and continuous on $\partial D$. We assume that the modulus of continuity $\omega(t)$ satisfyes the relation $$ \int\limits_0^x \frac{\omega(t)}{t} dt + x \int\limits_x^\infty \frac{\omega(t)}{t^2} dt \leq c\omega(x). $$ Let a function $\Phi$ map conformally the domain $\mathbb{C} \setminus D$ onto $\mathbb{C} \setminus \mathbb{D}$ with the normalization $\Phi(\infty) = \infty, \Phi^{\prime}(\infty) > 0$. We put $L_{1+t} = \{z \in \mathbb{C} \setminus D: |\Phi(z)| = 1+t\}, \delta_n(z) = \mathrm{dist} (z, L_{1+\frac{1}{n}}), z \in \partial D$. The main result of the paper is the following statement. Theorem 1. Assume that there exists a sequence of polynomials $P_n(u, v)$, $\deg P_n \leq n$, such that $$ |f(z) - P_n(\mathfrak{P}(z), \mathfrak{P}^{\prime}(z))| \leq C \delta^{r}_n(z)\omega(\delta_{n}(z)), z \in \partial D. $$ $C$ is independent on $n$ and $z$. Then $f \in H^{r+\omega}(D)$.