On the solutions of generalized Cauchy–Riemann system

Abstract

It is well known that the first order elliptic system with two unknown functions of two real variables can be represented in the form of a single equation with one unknown function of the complex variable. The last equation is known as the generalized Cauchy–Riemann system. In the past extensive research has been done regarding the last system, having created a comprehensive theory for such systems. But in the majority of research the last system was studied for bounded domains of the complex plane. As for unbounded domains, the theory for such systems is far from being complete. Therefore, it seems reasonable to study such systems in unbounded domains. In this regard, this paper is an attempt to study some properties of the solutions of such systems on the whole complex plane. The paper, mainly deals with the question of existence of the solutions of the generalized Cauchy–Riemann system in function spaces, in particular, in the space of bounded analytic functions. Without summability assumption on the coefficients of the equation, existence of the solutions for the whole complex plane is proved.

Keywords:

• analytic function
• Cauchy–Riemann system
• elliptic equation
• generalized analytic function
• spectral radius

AMS Subject Classifications:

• 35J45
• 35J46
• 30G20

Acknowledgments

The author thanks Professor E.M. Muhamadiev for setting the problem, giving the idea of the solution and for his continuous help to complete the paper and Professor A.A. Naimov for his help in Section 4, Theorem 4.6. The author would like to express his sincere gratitude to the Associate Editor and two anonymous referees for their useful comments, remarks and suggestions to improve the quality of the paper.