WEIGHTED NORM INEQUALITIES FOR FLAG SINGULAR INTEGRALS ON HOMOGENEOUS GROUPS

Abstract

Let $G$ be a homogeneous nilpotent Lie group. In this paper, we introduce a new class of multiparameter weights $A^{\mathcal{F}}_p$ associated with a flag $\mathcal{F}$ on $G$ and show that such class of weights can be characterized via two type of flag maximal operators. We then prove that singular integrals with flag kernels are bounded on $L^p_w(G)$, $1 \lt p \lt \infty$, when $w \in A_{p}^{\mathcal{F}}(G)$, which extends a recent result of Nagel-Stein-Wainger in [13]. As an application, we get weighted norm inequalities for the multiparameter Marcinkiewicz multipliers on Heisenberg groups introduced in [11].