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

Abstract

For f ∈ H p ( δ 2 ) f \in H^p(\delta ^2) , 0 > p ≤ 2 0>p\leq 2 , with Haar expansion f = ∑ f I × J h I × J f=\sum f_{I \times J}h_{I\times J} we constructively determine the Pietsch measure of the 2 2 -summing multiplication operator \[ M f : ℓ ∞ → H p ( δ 2 ) , ( φ I × J ) ↦ ∑ φ I × J f I × J h I × J . \mathcal {M}_f:\ell ^{\infty } \rightarrow H^p(\delta ^2), \quad (\varphi _{I\times J}) \mapsto \sum \varphi _{I\times J}f_{I \times J}h_{I \times J}. \] Our method yields a constructive proof of Pisier’s decomposition of f ∈ H p ( δ 2 ) f \in H^p(\delta ^2) \[ | f | = | x | 1 − θ | y | θ  and  ‖ x ‖ X 0 1 − θ ‖ y ‖ H 2 ( δ 2 ) θ ≤ C ‖ f ‖ H p ( δ 2 ) , |f|=|x|^{1-\theta }|y|^{\theta }\quad \quad \text { and }\quad \quad \|x\|_{X_0}^{1-\theta }\|y\|^{\theta }_{H^2(\delta ^2)}\leq C\|f\|_{H^p(\delta ^2)}, \] where X 0 X_0 is Pisier’s extrapolation lattice associated to H p ( δ 2 ) H^p(\delta ^2) and H 2 ( δ 2 ) H^2(\delta ^2) . Our construction of the Pietsch measure for the multiplication operator M f \mathcal {M}_f involves the Haar coefficients of f f and its atomic decomposition. We treated the one-parameter H p H^p -spaces in Houston Journal Math. 41 (2015), 639–668.