Two-dimensional Hardy-Littlewood theorem for functions with general monotone Fourier coefficients (Q6082534)

Let \(\{\phi_n(x)\}\) be an orthonormal system on \([a, b]\) such that \(|\phi_n(x)|\leq M\) for all \(x\in [a, b]\) and \(n\in \mathbb{N}\). \textit{R. E. A. C. Paley} [Stud. Math. 3, 226--238 (1931; Zbl 0003.35201)] proved that if \(p\in (1, 2]\), then for any \(f\in L_p(a, b)\) with Fourier coefficients \(\{c_n\}\) there holds \[ \sum_{n=1}^{\infty}|c_n|^p n^{p-2}\leq \mathrm{const} (p,M) \|f\|_p^p. \tag{1} \] In addition, if \(p\in [2,\infty)\), then for any sequence \(\{c_n\}\) with \[ \sum_{n=1}^{\infty}|c_n|^p n^{p-2}<\infty \] there exists a function \(f\in L_p(a, b)\) that has \(\{c_n\}\) as its Fourier coefficients and \[ \sum_{n=1}^{\infty}|c_n|^p n^{p-2}\geq \mathrm{const} (p,M) \|f\|_p^p. \tag{2} \] \textit{G. H. Hardy} and \textit{J. E. Littlewood} [J. Lond. Math. Soc. 6, 3--9 (1931; Zbl 0001.13504)] showed that if we restrict ourselves to sine or cosine series with monotone tending to zero coefficients, then both relations (1) and (2) hold for all \(p\in (1,\infty)\). Further research in this area has been carried out by several authors, including generalizations to weighted spaces \(L_{w(p,g)}^{q}\) and double Fourier series. At the same time, previously only nonnegative sequences of coefficients of certain classes were considered. The main purpose of this article is to show that for some kinds of double sequences, it is possible to prove an analogue of the Hardy-Littlewood theorem without restricting ourselves only to positive sequences.