A priori estimates of the positive real or imaginary part of a generalized analytic function

We denote by D=Dz={z:|z| 0, then U(z) ≥ K for all z∈D. The subject of this work is the generalization of this property to the real (imaginary) part of the solution to the elliptic system on D: ∂z¯w-q1(z)∂zw-q2(z)∂z¯w +A(z)w+B(z)w=0, where w=w(z)=u(z)+iv(z) is a desired complex function. ∂z¯=12(∂∂x+i∂∂y), ∂z=12(∂∂x-i∂∂y), are derivatives in Sobolev sense; q1(z) and q2(z) are given measurable complex functions satisfying the uniform ellipticity condition of the system |q1(z)|+|q2(z)|≤q0=const 2, also are given complex functions.