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 Houston Journal Math. 41 (2015), 639–668.