Bilinear forms on 𝐻^{∞} and bounded bianalytic functions

Abstract

Given an arbitrary Radon probability measure on the circle π \pi , a generlization of the classical Cauchy transform is obtained. These projections are used to prove that each bounded linear operator from a reflexive subspace of L 1 {L^1} or L 1 ( π ) / H 1 {L^1}(\pi )/{H^1} into H ∞ ( D ) {H^\infty }(D) admits a bounded extension. These facts lead to different variants of the cotype- 2 2 inequality for L 1 ( π ) / H 1 {L^1}(\pi )/{H^1} . Applications are given to absolutely summing operators and the existence of certain bounded bianalytic functions. For instance, we derive the Hilbert space factorization of arbitrary bounded linear operators from H ∞ ( D ) {H^\infty }(D) into its dual without an a priori approximation hypothesis, thus completing some of the work in [1]. Our methods give new information about the Fourier coefficients of H ∞ ( D × D ) {H^\infty }(D \times D) -functions, thus improving a theorem in [6].