On a natural connection between the entropy spaces and Hardy space

Author:
Romuald Dąbrowski

Journal:
Proc. Amer. Math. Soc. **104** (1988), 812-818

MSC:
Primary 42A45; Secondary 42B30, 46E99

DOI:
https://doi.org/10.1090/S0002-9939-1988-0964862-6

MathSciNet review:
964862

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Abstract | References | Similar Articles | Additional Information

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 [**5, 6**]. In fact, these operators define multiplier type bounded embeddings of entropy spaces to . 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 is a continuous periodic function of bounded Shannon entropy norm (resp. of bounded Lipschitz entropy norm with exponent ), and are the Fourier coefficients of , then (resp. ). 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 is a direct limit of the entropy spaces in an appropriate sense.

**[1]**Ronald R. Coifman,*A real variable characterization of 𝐻^{𝑝}*, Studia Math.**51**(1974), 269–274. MR**0358318**, https://doi.org/10.4064/sm-51-3-269-274**[2]**Romuald Dąbrowski,*Probability measure representation of norms associated with the notion of entropy*, Proc. Amer. Math. Soc.**90**(1984), no. 2, 263–268. MR**727246**, https://doi.org/10.1090/S0002-9939-1984-0727246-8**[3]**-,*Bochner integral and continuous functions of bounded entropy*, Preprint, 1986.**[4]**Romuald Dąbrowski,*On Fourier coefficients of a continuous periodic function of bounded entropy norm*, Bull. Amer. Math. Soc. (N.S.)**18**(1988), no. 1, 49–51. MR**919659**, https://doi.org/10.1090/S0273-0979-1988-15594-2**[5]**Peter L. Duren,*Theory of 𝐻^{𝑝} spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655****[6]**Paul Koosis,*Introduction to 𝐻_{𝑝} spaces*, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR**565451****[7]**Boris Korenblum,*On a class of Banach spaces of functions associated with the notion of entropy*, Trans. Amer. Math. Soc.**290**(1985), no. 2, 527–553. MR**792810**, https://doi.org/10.1090/S0002-9947-1985-0792810-2**[8]**B. Korenblum,*A generalization of two classical convergence tests for Fourier series, and some new Banach spaces of functions*, Bull. Amer. Math. Soc. (N.S.)**9**(1983), no. 2, 215–218. MR**707960**, https://doi.org/10.1090/S0273-0979-1983-15160-1**[9]**C. Sunberg,*Truncations of***B.M.O.***functions*, Indiana Univ. Math. J.**33**(1984).**[10]**Kôsaku Yosida,*Functional analysis*, 4th ed., Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 123. MR**0350358**

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0964862-6

Article copyright:
© Copyright 1988
American Mathematical Society