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Class of hypocomplex structures on the two-dimensional torus. (English) Zbl 1421.35050
Summary: We study the Hölder solvability of a class of complex vector fields on the torus \(\mathbb{T}^2\). We make use of the Theta function to associate a Cauchy-Pompeiu type integral operator. A similarity principle for the solutions of the equation \(Lu=au+b\bar{u}\) is obtained.

35F05 Linear first-order PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35C15 Integral representations of solutions to PDEs
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[1] Berhanu, Shiferaw; Cordaro, Paulo D.; Hounie, Jorge, An introduction to involutive structures, New Mathematical Monographs 6, xii+392 pp., (2008), Cambridge University Press, Cambridge · Zbl 1151.35011
[2] Bergamasco, Adalberto P.; Dattori da Silva, Paulo L.; Gonzalez, Rafael B., Existence and regularity of periodic solutions to certain first-order partial differential equations, J. Fourier Anal. Appl., 23, 1, 65-90, (2017) · Zbl 1357.35020
[3] Bergamasco, Adalberto P.; Cordaro, Paulo D.; Malagutti, Pedro A., Globally hypoelliptic systems of vector fields, J. Funct. Anal., 114, 2, 267-285, (1993) · Zbl 0777.58041
[4] Bergamasco, Adalberto; Parmeggiani, Alberto; Zani, S\'ergio; Zugliani, Giuliano, Classes of globally solvable involutive systems, J. Pseudo-Differ. Oper. Appl., 8, 4, 551-583, (2017) · Zbl 1382.58018
[5] Campana, C.; Dattori da Silva, P. L.; Meziani, A., Properties of solutions of a class of hypocomplex vector fields. Analysis and geometry in several complex variables, Contemp. Math. 681, 29-50, (2017), Amer. Math. Soc., Providence, RI · Zbl 1362.35088
[6] Campana, C.; Dattori da Silva, P. L.; Meziani, A., Riemann-Hilbert problem for a class of hypocomplex vector fields, Complex Var. Elliptic Equ., 61, 12, 1656-1667, (2016) · Zbl 1375.35293
[7] Dattori da Silva, P. L.; Meziani, A., Cohomology relative to a system of closed forms on the torus, Math. Nachr., 289, 17-18, 2147-2158, (2016) · Zbl 1368.58011
[8] Hounie, Jorge; Zugliani, Giuliano, Global solvability of real analytic involutive systems on compact manifolds, Math. Ann., 369, 3-4, 1177-1209, (2017) · Zbl 1380.35129
[9] Meziani, Abdelhamid, On first and second order planar elliptic equations with degeneracies, Mem. Amer. Math. Soc., 217, 1019, vi+77 pp., (2012) · Zbl 1250.35114
[10] Meziani, Abdelhamid, Representation of solutions of a singular Cauchy-Riemann equation in the plane, Complex Var. Elliptic Equ., 53, 12, 1111-1130, (2008) · Zbl 1162.30029
[11] Meziani, Abdelhamid, Hypoellipticity of nonsingular closed 1-forms on compact manifolds, Comm. Partial Differential Equations, 27, 7-8, 1255-1269, (2002) · Zbl 1017.58014
[12] Narasimhan, Raghavan, Compact Riemann surfaces, Lectures in Mathematics ETH Z\`‘urich, iv+120 pp., (1992), Birkh\'’auser Verlag, Basel · Zbl 0758.30002
[13] Rodin, Yuri L., Generalized analytic functions on Riemann surfaces, Lecture Notes in Mathematics 1288, vi+128 pp., (1987), Springer-Verlag, Berlin · Zbl 0637.30041
[14] Tr\`eves, Fran\ccois, Hypo-analytic structures, Princeton Mathematical Series 40, xviii+497 pp., (1992), Princeton University Press, Princeton, NJ · Zbl 0787.35003
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