Decomposition theorems for Hardy spaces on convex domains of finite type
The authors study the holomorphic Hardy spaces \({\mathcal H}^p(\omega)\), where \(\omega\) is a smoothly bounded convex domain of finite type in \({\mathbb C}^n\). They show that for \(0<p\leq 1\), \({\mathcal H}^p(\omega)\) admits an atomic decomposition. More precisely, they prove that each \(f\in {\mathcal H}^p(\omega)\) can be written as \(f = P_s(\sum_{j=0}^\infty \lambda_ja_j) = \sum_{j=0}^\infty \lambda_jP_s(a_j)\), where \(P_s\) is the Szegő projection, the \(a_j\)'s are real variable \(p\)-atoms on the boundary \(\partial\omega\), and the coefficients \(\lambda_j\) satisfy the condition \(\sum_{j=0}^\infty |\lambda_j|^p \leq C\|f\|^p_{\mathcal H^p}\). Moreover, they prove the following factorization theorem. Each \(f\in {\mathcal H}^p(\omega)\) can be written as \(f=\sum^\infty_{j=0}f_jg_j\), where \(f_j \in {\mathcal H}^{2p}(\omega)\), \(g_j \in {\mathcal H}^{2p}\omega\), and \(\sum^\infty_{j=0}\|f_j\|_{\mathcal H^{2p}}\|g_j\|_{\mathcal H^{2p}}\leq C\|f\|_{{\mathcal H}^p}\). Finally they extend these theorems to a class of domains of finite type that includes the strongly pseudoconvex domains and the convex domains of finite type.
scientific article; zbMATH DE number 1839943