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Weighted spaces of functions in the theory of generalized Cauchy-Riemann equations. (Russian. English summary) Zbl 1240.30206
The weighted functional spaces connected with generalized Cauchy-Riemann equations with singular coefficients are studied. Connections with other spaces are established and conjugate spaces are described. A foundational work in this direction is the monograph [I. N. Vekua, Обобщенные аналитические функции. Moskva: Nauka (1988; Zbl 0698.47036)], where the theory of equations of the form $\partial_{\bar{z}}w(z)+A(z)w(z)+B(z)\bar{w}(z)=0, \quad z\in G, \tag{1}$ is established; here $$G$$ is a bounded domain and $$A$$ and $$B$$ are given functions. In the article [M. Reissig and A. Timofeev, Complex Variables, Theory Appl. 50, No. 7–11, 653–672 (2005; Zbl 1084.30054)], the Dirichlet problem for such equations with $$G$$ being the unit disk and $$A\equiv0$$ was investigated. The novelty of this work consists in the fact that allowing a singularity in $$z=0$$ makes the coefficients $$B$$ belong to the weighted space $$S_{p}(G)$$, which is the union of the space $s_{p}(G)=\{B(z)| \sup_{\bar{G}}(| B(z)| p(| z| ))<+\infty\}$ and the set $$P$$ of functions $$p$$ possessing sufficiently general properties. The space $$S_{p}(G)$$ consists of functions $$f$$ in $$G$$, for which there exists a function $$p\in P$$, such that $$f(z)\in s_{p}(G)$$. Here, the functions $$p$$ are positive and nondecreasing on some semi-interval $$(0, t_{p}]$$, $$t_{_{p}}>1$$, and such that $$\lim\limits_{t\rightarrow+0}p(t)=0$$, $$\int_{0}^{t_{p}}\frac{dt}{p(t)}<+\infty$$. In the article under review, the author extends the study of functions from the class $$P$$.

##### MSC:
 30G20 Generalizations of Bers and Vekua type (pseudoanalytic, $$p$$-analytic, etc.)
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