Abstract

We establish a weighted Hardy space theory associated with flag structures. This theory differs from those in the classical one-parameter and the product settings, and includes weighted Hardy spaces $H_{\mathcal{ℱ},w}^p$, weighted Carleson measure spaces ${\mathit{CMO}}_{\mathcal{ℱ},w}^p$ (the dual spaces of $H_{\mathcal{ℱ},w}^p$), and the boundedness of singular integrals with flag kernels on these spaces. We also derive a Calderón–Zygmund decomposition and provide interpolation of operators acting on $H_{\mathcal{ℱ},w}^p$. Examples and counterexamples are constructed to clarify the relations between classes of one-parameter, product and flag $A_p$ weights. The main tool for our approach is the weighted Littlewood–Paley–Stein theory associated with the flag structure.

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Mathematical Subject Classification 2010

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