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

For $f\in H^p({\delta }^2)$, $0<p\le 2$, with Haar expansion $f=\sum f_{I \times J}h_{I\times J}$weconstructivelydeterminethePietschmeasureofthe$2$-summing multiplication operator $${\mathcal{M}}_f:{\mathrm{\ell }}^{\mathrm{\infty }}\to H^p({\delta }^2),({\phi }_{I\times J})\mapsto \sum {\phi }_{I\times J}{f}_{I\times J}{h}_{I\times J}\mathrm{.}$$ Our method yields a constructive proof of Pisier’s decomposition of $f \in H^p(\delta^2)$ $$\mathrm{\mid }f\mathrm{\mid }=\mathrm{\mid }x{\mathrm{\mid }}^{1-\theta }\mathrm{\mid }y{\mathrm{\mid }}^{\theta }\text{ and }\mathrm{\parallel }x{\mathrm{\parallel }}_{X_0}^{1-\theta }\mathrm{\parallel }y{\mathrm{\parallel }}_{H^2({\delta }^2)}^{\theta }\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{H^p({\delta }^2)},$$ 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.].