Journal of Operator Theory

Volume 44, Issue 1, Summer 2000 pp. 63-90.

Bilinear operators on homogeneous groups

Authors: Loukas Grafakos (1), and Xinwei Li (2)
Author institution: (1) Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
(2) Department of Mathematics, Washington University, St. Louis, MO 63130, USA

Summary: Let $H^p$ denote the Lebesgue space $L^p$ for $p>1$ and the Hardy space $H^p$ for $p\le 1$. For $0<p,q,r<\infty$, we study $H^p \times H^q \rightarrow H^r$ mapping properties of bilinear operators given by finite sums of products of Calder\' on-Zygmund operators on stratified homogeneous Lie groups. When $r\le 1$, we show that such mapping properties hold when a number of moments of the operator vanish. This hypothesis is natural and the conditions imposed are the minimal required for any operator of this type to map into the space $H^r$. Our proofs employ both the maximal function and atomic characterization of $H^p$. We also discuss some applications.

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