Vol. 302, No. 2, 2019

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Boundedness of singular integrals with flag kernels on weighted flag Hardy spaces

Yongsheng Han, Chin-Cheng Lin and Xinfeng Wu

Vol. 302 (2019), No. 2, 545–598
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 ${H}_{\mathsc{ℱ},w}^{p}$, weighted Carleson measure spaces ${\mathit{CMO}}_{\mathsc{ℱ},w}^{p}$ (the dual spaces of ${H}_{\mathsc{ℱ},w}^{p}$), 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 ${H}_{\mathsc{ℱ},w}^{p}$. 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.

Keywords
Calderón–Zygmund decomposition, duality, interpolation, singular integrals with flag kernels, weighted flag Carleson measure spaces, weighted flag Hardy spaces
Primary: 42B20
Secondary: 42B30