In this paper, conjugate $k$-holomorphic functions and generalized $k$-holomorphic functions are defined in the two-dimensional complex space, and the corresponding Riemann boundary value problems and the inverse problems are discussed on generalized bicylinders. By the characteristics of the corresponding functions and boundary properties of the Cauchy type singular integral operators with conjugate $k$-holomorphic kernels, the general solutions and special solutions of the corresponding boundary value problems are studied in a detailed fashion, and the integral expressions of the solutions are obtained.
Yanyan CUI
,
Chaojun WANG
,
Yonghong XIE
,
Yuying QIAO
. BOUNDARY VALUE PROBLEMS OF CONJUGATE AND GENERALIZED $k$-HOLOMORPHIC FUNCTIONS IN $\mathbb{C}^2$*[J]. Acta mathematica scientia, Series B, 2024
, 44(5)
: 1837
-1852
.
DOI: 10.1007/s10473-024-0511-6
[1] Balk M B. Polyanalytic Functions. Berlin: Akademie Verlay, 1991
[2] Youssfi E H. Polyanalytic reproducing kernels in $C^n$. Complex Anal Synerg, 2021, 7: 1-7
[3] Ghiloni R. Slice-by-slice and global smoothness of slice regular and polyanalytic functions. Annali di Matematica, 2022, 201: 2549-2573
[4] Vasilevski N. On the polyanalytic and anti-polyanalytic function spaces. J Math Sci, 2022, 266: 210-230
[5] El Gourari A, Ghanmi A, Rouchdi I. Bicomplex polyharmonicity and polyholomorphy. Complex Anal Oper Theory, 2022, 16: Art 67
[6] Abkar A. Mean approximation in Bergman spaces of polyanalytic functions. Anal Math Phys, 2022, 12: Art 52
[7] Abdulhadi Z, Hajj L E. On the univalence of poly-analytic functions. Comput Meth Funct Th, 2021, 22(1): 169-181
[8] Alpay D, Colombo F, Diki K, et al. Reproducing kernel hilbert spaces of polyanalytic functions of infinite order. Integr Equ Oper Theory, 2022, 94: Art 98
[9] Qiao Y, Cui Y, Li Z, Wang L. $k$-holomorphic functions in spaces of several complex variables. Complex Variables and Elliptic Equations, 2020, 65(11): 1826-1845
[10] Çelebi A O, Mityushev V V. Schwarz problem for bi-analytic functions in multiply connected domains. Complex Var Elliptic Equ, 2015, 60: 566-582
[11] Guo G, Jin C, Dang P, et al. Riemann-Hilbert problems with shift on the Lyapunov curve for null-solutions of iterated Beltrami equations. Bound Value Probl, 2019, 98: 1-34
[12] Bory-Reyes J, Abreu-Blaya R, Pérez-de la Rosa M A, et al. Hilbert and Poincaré-Bertrand formulas in polyanalytic function theory involving higher order lipschitz classes. Complex Anal Oper Th, 2021, 15: Art 94
[13] Begehr H, Shupeyeva B. Polyanalytic boundary value problems for planar domains with harmonic Green function. Anal Math Phys, 2021, 11: Art 137
[14] Li Y, Sun H. A note on the Julia sets of entire solutions to delay differential equations. Acta Math Sci, 2023, 43B(1): 143-155
[15] Bradley P E. Generalised Diffusion on Moduli Spaces of p-Adic Mumford Curves. P-Adic Num Ultrametr Anal Appl, 2020, 12: 73-89
[16] Valencia D E G, Blaya R A, Alejandre M P A, et al. On the Riemann problem in fractal elastic media. Anal Math Phys, 2023, 13: Art 3
[17] Wu X. Some oscillation criteria for a class of higher order nonlinear dynamic equations with a delay argument on time scales. Acta Mathematica Scientia, 2021, 41B(5): 1474-1492
[18] Huang S.A nonlinear boundary value problem for analytic functions of several complex variables
(in Chinese). Acta Math Sci, 1997, 17A: 382-388
[19] Yang P. The $k$-regular function and some boundary value problems. Journal of Sichuan Normal University (Natural Science), 2001, 24: 5-8
[20] Bekya N H. Generalized Analytic Functions.Beijing: People's Education Press, 1960
