Abstract
This work consists of two sections. The first section examines the elliptic system of equations ∂z¯u+a(z)u+b(z)u¯=f. The concept of Vekua spaces (V-spaces) is introduced, namely: a Banach function space B is a Vekua space if the theorem about continuity of the solution of an inhomogeneous equation and the theorem about the representation of solutions of a homogeneous equation holds for any functions a(z),b(z) and f(z). The following main statement is obtained (Theorem 7.1): A function space B possessing the properties 1∘−3∘ of § 2 is a V – space if and only if B↪P1 (for the definition of P1 see § 2). From this result it follows that a symmetric space is a V space if and only if it is continuously embedded in the Lorentz space L(2,1) (Theorem 7.3). Thus, we can say that the widest space to which the Vekua theory can be extended is P1(⋅), and among all symmetric spaces - L(2,1). In the second section, we study the following equation with singular point ∂z¯u+a(z)|z|u+b(z)|z|u¯=f. The paper proves the existence of solutions to this equation in the functional class B introduced in the first part of the article. Representations of the 1st and 2nd types of solutions from this class are obtained.