ON THE CAUCHY TRANSFORM OF FUNCTIONALS ON A BERGMAN SPACE

The strong dual space of the Bergman space B-2(G) = {f is an element o f H(G):parallel to f parallel to(B2(G))(2) = integral(G) \f(x)\(2) dv( z) < infinity} is described in terms of the Cauchy transformation, whe re v(z) is Lebesgue measure and G is a simply connected domain with bo undary of class C1+0. As a normed space, B-2(G) is isomorphic to the space B-2(1)(C\(G) over bar = {gamma(zeta) is an element of H(C\(G) ov er bar), gamma(infinity) = 0:parallel to gamma parallel to(B21(C\(G) o ver bar)) = integral C\(G) over bar\gamma'(zeta)\(2) dv(zeta) < infini ty}. An example is given of a domain with nonsmooth boundary for which the spaces B-2(G) and B-2(1)(C\(G) over bar) are not isomorphic.