The space K of all Cauchy transforms becomes a Banach space under the norm ‖f‖_K = inf ‖μ‖_M, where the infimum is taken over all Borel measures μ satisfying (1.1). The Banach space K is clearly the quotient of the Banach space M of Borel measures by the subspace of measures with vanishing Cauchy transforms. It is an immediate consequence of the F. and M. Riesz theorem that a Borel measure μ has a vanishing Cauchy transform if and only if μ has the form dμ = f dm, where f ∈ H_0~1 and m is normalized Lebesgue measure on T.
展开▼
