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.

We costruct functions in $H_w^1$ ($w\in A_1$) whose Fourier integral expansions are almost everywhere non-summable with respect to the Bochner-Riesz means of the critical order.

We study Hardy, Bergman, Bloch, and BMO spaces on convex domains of finite type in $n$-dimensional complex space. Duals of these spaces are computed. The essential features of complex domains of finite type, that make these theorems possible, are isolated.

We give several new characterizations of the dual of the dyadic Hardy space H1,d(T2), the so-called dyadic BMO space in two variables and denoted BMOdprod. These include characterizations in terms of Haar multipliers, in terms of the "symmetrised paraproduct" Λb, in terms of the rectangular BMO norms of the iterated "sweeps", and in terms of nested commutators with dyadic martingale transforms. We further explore the connection between BMOdprod and John-Nirenberg type inequalities, and study a scale...