Absolutely summing operators and atomic decomposition in bi-parameter Hardy spaces

Abstract

For $f \in H^p(\delta^2)$, $0<p\leq 2$, with Haar expansion $f=\sum f_{I \times J}h_{I\times J}$ we constructively determine the Pietsch measure of the $2$-summing multiplication operator \[\mathcal{M}_f:\ell^{\infty} \rightarrow H^p(\delta^2), \quad (\varphi_{I\times J}) \mapsto \sum \varphi_{I\times J}f_{I \times J}h_{I \times J}. \] Our method yields a constructive proof of Pisier's decomposition of $f \in H^p(\delta^2)$ \[|f|=|x|^{1-\theta}|y|^{\theta}\quad\quad \text{ and }\quad\quad \|x\|_{X_0}^{1-\theta}\|y\|^{\theta}_{H^2(\delta^2)}\leq C\|f\|_{H^p(\delta^2)}, \] where $X_0$ is Pisier's extrapolation lattice associated to $H^p(\delta^2)$ and $H^2(\delta^2)$. Our construction of the Pietsch measure for the multiplication operator $\mathcal{M}_f$ involves the Haar coefficients of $f$ and its atomic decomposition. We treated the one-parameter $H^p$-spaces in [P.F.X M\"uller, J.Penteker, $p$-summing multiplication operators, dyadic Hardy spaces and atomic decomposition, Houston Journal Math.,41(2):639-668,2015.].