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Transactions of the American Mathematical Society

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Embedding $ L\sp{1}$ in $ L\sp{1}/H\sp{1}$


Author: J. Bourgain
Journal: Trans. Amer. Math. Soc. 278 (1983), 689-702
MSC: Primary 46E30; Secondary 42A99, 46B25, 60G46
DOI: https://doi.org/10.1090/S0002-9947-1983-0701518-9
MathSciNet review: 701518
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Abstract: It is proved that $ {L^1}$ is isomorphic to a subspace of $ {L^1}/{H^1}$. More precisely, there exists a diffuse $ \sigma $-algebra $ \mathfrak{S}$ on the circle such that the corresponding expectation $ {\mathbf{E}}:{H^\infty } \to {L^\infty }({\mathbf{C}})$ is onto. The method consists in studying certain martingales on the product $ {\prod ^{\mathbf{N}}}$.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0701518-9
Article copyright: © Copyright 1983 American Mathematical Society

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