Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: J. Bourgain
Journal: Trans. Amer. Math. Soc. 286 (1984), 313-337
MSC: Primary 46J15; Secondary 46E15, 47B10
MathSciNet review: 756042
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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].

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46J15, 46E15, 47B10

Retrieve articles in all journals with MSC: 46J15, 46E15, 47B10

Additional Information

Keywords: Bounded analytic function, linear operator factorization
Article copyright: © Copyright 1984 American Mathematical Society