Boundedness of paraproducts on spaces of homogeneous type II

Abstract

This article is the second part of two works by the same authors on the same topic. Let $(\mathcal{X},d,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors introduce the notion of paraproducts on $\mathcal{X}$ and obtain their boundedness from $H^p\left(\mathcal{X}\right)\times H^q\left(\mathcal{X}\right)$ into $H^r\left(\mathcal{X}\right)$ for any given $p,q,r\in \left(n/\right(n+\eta ),\mathrm{\infty })$ satisfying $1/r=1/p+1/q$, where $H^p\left(\mathcal{X}\right)$, for any given $p\in \left(n/\right(n+\eta ),\mathrm{\infty })$, denotes the Hardy space on $\mathcal{X}$. Moreover, the authors also establish the endpoint boundedness of these paraproducts on $\mathcal{X}$ when p or q is ∞ or 1 with $H^p\left(\mathcal{X}\right)$ or $H^q\left(\mathcal{X}\right)$ replaced by $\mathrm{B}\mathrm{M}\mathrm{O}\left(\mathcal{X}\right)$, $L^{\mathrm{\infty }}\left(\mathcal{X}\right)$ or $L^1\left(\mathcal{X}\right)$, and hence give a complete picture on the boundedness of paraproducts on $\mathcal{X}$. The main novelty of this article is the avoidance of dependence on the reverse doubling assumption of the considered measure μ, which is achieved by fully using the geometrical properties of $\mathcal{X}$ expressed via its equipped quasi-metric d, dyadic reference points and dyadic cubes.

Keywords:

• Space of homogeneous type
• paraproduct
• Hardy space
• bilinear estimate

2010 Mathematics Subject Classifications:

• Primary 42B20
• Secondary 42B30
• 47A07
• 42C40
• 46E36
• 30L99

Acknowledgements

The second author would like to express his deep thanks to Ziyi He for a helpful discussion on the proof of Theorem 2.1.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This project is supported by the National Natural Science Foundation of China (Grant Nos. 11701160, 11871100, 11971058, 11761131002 and 11671185) and Der-Chen Chang is partially supported by the National Science Foundation (NSF) (USA) (Grant No.  DMS-1408839) and a McDevitt Endowment Fund at Georgetown University (Grant No. 2016-2026).