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Transactions of the American Mathematical Society

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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
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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].


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DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0756042-5
Keywords: Bounded analytic function, linear operator factorization
Article copyright: © Copyright 1984 American Mathematical Society