On the Cauchy transform of weighted Bergman spaces

The problem of describing the range of a Bergman space B_2(G) under the Cauchy transform K for a Jordan domain G was solved by Napalkov (Jr) and Yulmukhametov. It turned out that K(B_2(G))=B_2^1(C\bar G) if and only if G is a quasidisk; here B_2^1(C\bar G) is the Dirichlet space of the complement of \bar G. The description of K(B_2(G)) for an integrable Jordan domain is given in [S. Merenkov, "On the Cauchy transform of the Bergman space", Mat. Fiz. Anal. Geom., 7 (2000), no. 1, 119-127]. In the present paper we give a description of K(B_2(G,\omega)) analogous to the one given in [S. Merenkov, "On the Cauchy transform of the Bergman space", Mat. Fiz. Anal. Geom., 7 (2000), no. 1, 119-127] for a weighted Bergman space B_2(G,\omega) with a weight \omega\ which is constant on level lines of the Green function of G. In the case G=D, the unit disk, and under some additional conditions on the weight \omega, K(B_2(D,\omega))=B_2^1(C\bar{D}, \omega^{-1}), a weighted analogue of a Dirichlet space.