We study the holomorphic Hardy-Orlicz spaces ${}^{\Phi }\left(\Omega \right)$, where Ω is the unit ball or, more generally, a convex domain of finite type or a strictly pseudoconvex domain in ℂⁿ. The function Φ is in particular such that $¹\left(\Omega \right){\subset }^{\Phi }\left(\Omega \right){\subset }^p\left(\Omega \right)$ for some p > 0. We develop maximal characterizations, atomic and molecular decompositions. We then prove weak factorization theorems involving the space BMOA(Ω). As a consequence, we characterize those Hankel operators which are bounded from ${}^{\Phi }\left(\Omega \right)$ into ¹(Ω).

This work is an introduction to anisotropic spaces of holomorphic functions, which have $\omega$-weight and are generalizations of Bloch spaces on a unit ball. We describe the holomorphic Bloch space in terms of the corresponding $L_{\omega }^{\infty }$ space. We establish a description of ${\left(A^p\left(\omega \right)\right)}^{*}$ via the Bloch classes for all $0<p\le 1$.

We construct a function f holomorphic in a balanced domain D in ${ℂ}^N$ such that for every positive-dimensional subspace Π of ${ℂ}^N$, and for every p with 1 ≤ p < ∞, ${f|}_{\Pi \cap D}$ is not $L^p$-integrable on Π ∩ D.