On a natural connection between the entropy spaces and Hardy space 𝑅𝑒𝐻¹

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 Re H 1 {\text {Re}}{H^1} [5, 6]. In fact, these operators define multiplier type bounded embeddings of entropy spaces to Re H 1 {\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 f is a continuous periodic function of bounded Shannon entropy norm (resp. of bounded Lipschitz entropy norm with exponent q , 0 > q > 1 q,0 > q > 1 ), and { a n } n ∈ Z {\left \{ {{a_n}} \right \}_{n \in {\mathbf {Z}}}} are the Fourier coefficients of f f , then ∑ n ≠ 0 | a n ( log(|n|)/n ) | > ∞ \sum \nolimits _{n \ne 0} {|{a_n}({\text {log(|n|)/n}})| > \infty } (resp. ∑ n ≠ 0 | a n | / |n| ) q > ∞ \sum \nolimits _{n \ne 0} {|{a_n}|/{\text {|n|}}{)^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 Re ⁡ H 1 \operatorname {Re} {H^1} is a direct limit of the entropy spaces in an appropriate sense.