The hardy and bellman operators in spaces connected with H($\mathbb{T}$) and BMO($\mathbb{T}$)

Abstract

Assume that 1 ≤ p < ∞ and a function f ∈ L ^p[0, π] has the Fourier series $\sum\limits_{n = 1}^\infty {a_n }$ cos nx. According to one result of G.H. Hardy, the series $\sum\limits_{n = 1}^\infty {n^{ - 1} } \sum\limits_{k = 1}^n {a_k }$ cos nx is the Fourier series for a certain function $\mathcal{H}$(f) ∈ L ^p[0, π]. But if 1 < p ≤ ∞ and f ∈ L ^p[0, π], then the series $\sum\limits_{n = 1}^\infty {\sum\limits_{k = n}^\infty {k^{ - 1} a_k } }$ cos nx is the Fourier series for a certain function $\mathcal{B}$(f) ∈ L ^p[0, π]. Similar assertions are true for sine series. This allows one to define the Hardy operator $\mathcal{H}$ on L ^p($\mathbb{T}$), 1 ≤ p < ∞, and to define the Bellman operator $\mathcal{B}$ on L ^p($\mathbb{T}$), 1 < p ≤ ∞. In this paper we prove that the Bellman operator boundedly acts in VMO($\mathbb{T}$), and the Hardy operator also maps a certain subspace C($\mathbb{T}$) onto VMO($\mathbb{T}$). We also prove the invariance of certain classes of functions with given majorants of modules of continuity or best approximations in the spaces H($\mathbb{T}$), L($\mathbb{T}$), VMO($\mathbb{T}$) with respect to the Hardy and Bellman operators.