On a natural connection between the entropy spaces and Hardy space $\textrm {Re} H^ 1$

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 $\sum \nolimits _{n \ne 0} {|{a_n}({\text {log(|n|)/n}})| < \infty }$ (resp. $\sum \nolimits _{n \ne 0} {|{a_n}|/{\text {|n|}}{)^q} < \infty }$). 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 $\operatorname {Re} {H^1}$ is a direct limit of the entropy spaces in an appropriate sense.