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

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 ^p _F,w, weighted Carleson measure spaces CMO^p _F,w (the dual spaces of H ^p _F,w), 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 ^p _F,w. 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.