## Illinois Journal of Mathematics

### 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 Szegö 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.

#### Article information

Source
Illinois J. Math., Volume 46, Number 1 (2002), 207-232.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258136151

Digital Object Identifier
doi:10.1215/ijm/1258136151

Mathematical Reviews number (MathSciNet)
MR1936086

Zentralblatt MATH identifier
1016.32005

#### Citation

Grellier, Sandrine; Peloso, Marco M. Decomposition theorems for Hardy spaces on convex domains of finite type. Illinois J. Math. 46 (2002), no. 1, 207--232. doi:10.1215/ijm/1258136151. https://projecteuclid.org/euclid.ijm/1258136151