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Multi-Parameter Hardy Spaces Theory and Endpoint Estimates for Multi-Parameter Singular Integrals
About this Title
Guozhen Lu, Jiawei Shen and Lu Zhang
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 281, Number 1388
ISBNs: 978-1-4704-5537-8 (print); 978-1-4704-7321-1 (online)
DOI: https://doi.org/10.1090/memo/1388
Published electronically: January 3, 2023
Keywords: Multi-parameter dilations,
singular Radon transforms,
Calderón-Zygmund operators,
discrete Littlewood-Paley-Stein theory,
multi-parameter Hardy spaces
Table of Contents
Chapters
- 1. Introduction
- 2. Single-parameter theory
- 3. Multi-parameter setting: Product theory
- 4. General multi-parameter singular integrals and Hardy spaces
Abstract
The main purpose of this paper is to establish the theory of the multi-parameter Hardy spaces $H^p$ ($0<p\leq 1$) associated to a class of multi-parameter singular integrals extensively studied in the recent book of B. Street (2014), where the $L^p$ $(1<p<\infty )$ estimates are proved for this class of singular integrals. This class of multi-parameter singular integrals are intrinsic to the underlying multi-parameter Carnot-Carathéodory geometry, where the quantitative Frobenius theorem was established by B. Street (2011), and are closely related to both the one-parameter and multi-parameter settings of singular Radon transforms considered by Stein and Street (2011, 2012a, 2012b, 2013).
More precisely, Street (2014) studied the $L^p$ $(1<p<\infty )$ boundedness, using elementary operators, of a type of generalized multi-parameter Calderón Zygmund operators on smooth and compact manifolds, which include a certain type of singular Radon transforms. In this work, we are interested in the endpoint estimates for the singular integral operators in both one and multi-parameter settings considered by Street (2014). Actually, using the discrete Littlewood-Paley-Stein analysis, we will introduce the Hardy space $H^p$ ($0<p\leq 1$) associated with the multi-parameter structures arising from the multi-parameter Carnot-Carathéodory metrics using the appropriate discrete Littlewood-Paley-Stein square functions, and then establish the Hardy space boundedness of singular integrals in both the single and multi-parameter settings. Our approach is much inspired by the work of Street (2014) where he introduced the notions of elementary operators so that the type of singular integrals under consideration can be decomposed into elementary operators.
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