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Representation of solutions of a singular Cauchy-Riemann equation in the plane. (English) Zbl 1162.30029
Motivated by the investigations of Z. D. Uzmanov [Complex Variables, Theory Appl. 26, No. 1–2, 41–52 (1994; Zbl 0990.30501)] the author analyzed a generalized Cauchy-Riemann equation of the type $\partial_{\overline z}=\frac{a(\theta)} {2\overline z}w+\frac{b(\theta)} {2\overline z}\overline w+F,$ where $$a,b$$ are $$2\pi$$-periodic functions and $$a$$ fulfils additionally the condition $\int^{2\pi}_0a(\theta)d\theta\in\mathbb{R}.$ A generalized Borel-Pompeiu formula with kernel functions $$\Omega_ii=1,2$$ is deduced and consequences are studied. On the unit circle a $$L_2$$-orthonormal complete system of eigenfunctions of a suitable equation is constructed. These eigenfunctions are used to get a representation of the kernels $$\Omega_ii=1,2$$ of the Borel-Pompeiu formula. Very interesting and highly non-trivial investigations of these kernels with spectral methods are lined out. A corresponding Teodorescu transform (T-operator) is defined and mapping properties are studied. The results can be transferred to slightly generalized equations of the above mentioned Cauchy-Riemann equation.

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