×

zbMATH — the first resource for mathematics

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.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.