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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On a natural connection between the entropy spaces and Hardy space $ {\rm Re}\,H\sp 1$


Author: Romuald Dąbrowski
Journal: Proc. Amer. Math. Soc. 104 (1988), 812-818
MSC: Primary 42A45; Secondary 42B30, 46E99
MathSciNet review: 964862
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Abstract: In 1983 B. Korenblum [7, 8] introduced a class of Banach function spaces associated with the notion of entropy (we will call these spaces and their norms entropy spaces and entropy norms, respectively). Entropy spaces were used in [8] as a tool for proving a new convergence test for Fourier series which includes classical tests of Dirichlet-Jordan and Dini-Lipschitz.

In this paper we construct natural linear operators from the entropy spaces to Hardy space $ {\text{Re}}{H^1}$ [5, 6]. In fact, these operators define multiplier type bounded embeddings of entropy spaces to $ {\text{Re}}{H^1}$. As a corollary we obtain a growth condition for Fourier coefficients of a continuous periodic function of bounded entropy norm (as announced in [4]). In particular, we show that if $ f$ is a continuous periodic function of bounded Shannon entropy norm (resp. of bounded Lipschitz entropy norm with exponent $ q,0 < q < 1$), and $ {\left\{ {{a_n}} \right\}_{n \in {\mathbf{Z}}}}$ are the Fourier coefficients of $ f$, then $ \sum\nolimits_{n \ne 0} {\vert{a_n}({\text{log(\vert n\vert)/n}})\vert < \infty } $ (resp. $ \sum\nolimits_{n \ne 0} {\vert{a_n}\vert/{\text{\vert n\vert}}{)^q} < \infty } $). In §4 we study the relationship between the dual spaces of entropy spaces and space B.M.O. of functions of bounded mean oscillation. In §5 we conjecture that $ \operatorname{Re} {H^1}$ is a direct limit of the entropy spaces in an appropriate sense.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0964862-6
PII: S 0002-9939(1988)0964862-6
Article copyright: © Copyright 1988 American Mathematical Society