Decomposition theorems for Hardy spaces on convex domains of finite type

Abstract

In this paper we study the holomorphic Hardy space $\mathcal{H}^p(\Omega)$, where $\Omega$ is a smoothly bounded convex domain of finite type in $\mathbb{C}^n$. We show that for $0<p\le1$, $\mathcal{H}^p(\Omega)$ admits an atomic decomposition. More precisely, we prove that each $f\in\mathcal{H}^p(\Omega)$ can be written as $f=P_S(\sum_{j=0}^{\infty}\nu_j a_j)=\sum_{j=0}^{\infty}\nu_j P_S(a_j)$, where $P_S$ is the Szego projection, the $a_j$'s are real variable $p$-atoms on the boundary $\partial\Omega$, and the coefficients $\nu_j$ satisfy the condition $\sum_{j=0}^{\infty}|\nu_j|^p \lesssim\|f\|_{\mathcal{H}^p(\Omega)}^p$. Moreover, we prove the following factorization theorem. Each $f\in\mathcal{H}^p(\Omega)$ can be written as $f=\sum_{j=0}^{\infty}f_j g_j$, where $f_j\in\mathcal{H}^{2p}$, $g_j\in\mathcal{H}^{2p}$, and $\sum_{j=0}^{\infty}\|f_j\|_{\mathcal{H}^{2p}} \|g_j\|_{\mathcal{H}^{2p}}$ $\lesssim\|f\|_{\mathcal{H}^p(\Omega)}$. Finally, we extend these theorems to a class of domains of finite type that includes the strongly pseudoconvex domains and the convex domains of finite type.