Bilinear forms on $H^{\infty }$ and bounded bianalytic functions

Abstract:

Given an arbitrary Radon probability measure on the circle $\pi$, 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 ${L^1}$ or ${L^1}(\pi )/{H^1}$ into ${H^\infty }(D)$ admits a bounded extension. These facts lead to different variants of the cotype-$2$ inequality for ${L^1}(\pi )/{H^1}$. 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 ${H^\infty }(D)$ 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 ${H^\infty }(D \times D)$-functions, thus improving a theorem in [6].