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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.

30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)
Full Text: DOI
[1] Bers L, Contr. Theory PDE, Ann. Math. Studies 33 pp 69– (1954)
[2] Vekua I, Generalized Analytic Functions (1962)
[3] DOI: 10.1080/17476939408814762 · Zbl 0990.30501 · doi:10.1080/17476939408814762
[4] Usmanov ZD, Pitman Monographs and Surveys in Pure and Applied Mathematics 85 (1997)
[5] DOI: 10.1080/0278107031000103106 · Zbl 1051.30044 · doi:10.1080/0278107031000103106
[6] DOI: 10.1080/0278107031000152599 · Zbl 1041.30022 · doi:10.1080/0278107031000152599
[7] Birkhoff G, Ordinary Differential Equations, 2. ed. (1969)
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