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The purpose of this paper is to obtain a discrete version for the Hardy spaces $H^p\left(\mathbb{Z}\right)$ of the weak factorization results obtained for the real Hardy spaces $H^p\left(ℝⁿ\right)$ by Coifman, Rochberg and Weiss for p > n/(n+1), and by Miyachi for p ≤ n/(n+1). It represents an extension, in the one-dimensional case, of the corresponding result by A. Uchiyama who obtained a factorization theorem in the general context of spaces X of homogeneous type, but with some restrictions on the measure that exclude the case of points...

We study various characterizations of the Hardy spaces $H^p\left(\mathbb{Z}\right)$ via the discrete Hilbert transform and via maximal and square functions. Finally, we present the equivalence with the classical atomic characterization of $H^p\left(\mathbb{Z}\right)$ given by Coifman and Weiss in [CW]. Our proofs are based on some results concerning functions of exponential type.