Inequalities for discrete Hardy spaces (Q5932744)

The Hardy space \(H^p(\mathbb R^n)\), \(0<p\leq 1\), has atomic decomposition and molecular characterization. Using these properties, one can obtain the boundedness of many operators and can study certain multiplier theorems. It is also well-known that the Hardy space \(H^p(\mathbb R^n)\), \(0<p<\infty\), can be defined by many equivalent definitions. For instance, on the one dimensional space \(\mathbb R^1\), \(f\in H^p(\mathbb R)\) if and only if \(\|f\|_{H^p}=\|f\|_{L^p}+\|Hf\|_{L^p}<\infty\), where \(Hf\) is the Hilbert transform of \(f\) which is defined by \[ Hf(x)=\text{p.v.} 1/\pi \int_{\mathbb R^1}(x-y)^{-1}f(y) dy. \] Consider the discrete space \(\mathbb Z=\{\text{all integers}\}\). For any fixed \(0<p<\infty\), the discrete Hardy space \(H^p(\mathbb Z)\) can be defined to be the space of all sequences \(c=\{c(n)\}_{n\in \mathbb Z}\) such that \(\|c\|_{H^p}+\|c\|_{\ell^p}+\|\mathcal H^d c\|_{\ell^p}<\infty\), where \(\mathcal H^d c\) is the discrete Hilbert transform of \(c\) given by \[ \mathcal H^d c(n)=\sum_{k\neq n} (n-k)^{-1}c(k),\quad n\in \mathbb Z. \] In an earlier paper, \textit{S. Boza} and \textit{M. J. Carro} [Stud. Math. 129, No. 1, 31-50 (1998; Zbl 0903.42011)] established the atomic decomposition of \(H^p(\mathbb Z)\). In this reviewed paper, based on the result of Boza and Carro, the authors show the molecular characterization of \(H^p(\mathbb Z)\). As applications, on the space \(H^p(\mathbb Z)\), they obtain the Marcinkiewicz multiplier theorem and an inequality on fractional integrals.