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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 \(P\).

30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)
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