Abstract
Assume that 1 ≤ p < ∞ and a function f ∈ Lp[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) ∈ Lp[0, π]. But if 1 < p ≤ ∞ and f ∈ Lp[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) ∈ Lp[0, π]. Similar assertions are true for sine series. This allows one to define the Hardy operator \( \mathcal{H} \) on Lp(\( \mathbb{T} \)), 1 ≤ p < ∞, and to define the Bellman operator \( \mathcal{B} \) on Lp(\( \mathbb{T} \)), 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.
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