Boundedness of paraproducts on spaces of homogeneous type I

Abstract

This article is the first 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 establish some basic estimates and a modified inhomogeneous discrete Calderón reproducing formula, which play essential roles in the proof of the boundedness of paraproducts in the second part of this work, and are also of independent interest. To escape the reverse doubling property of the equipped measure μ, the authors make full use of the geometrical properties of $\mathcal{X}$ expressed by its equipped quasi-metric d, dyadic reference points and dyadic cubes.

COMMUNICATED BY:

• R. P. Gilbert

Keywords:

• Space of homogeneous type
• Calderón reproducing formula
• Calderón–Zygmund operator
• wavelet

Mathematics Subject Classifications:

• Primary 42B20
• Secondary 42B25
• 42C40
• 30L99

Disclosure statement

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

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

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. 2016-2016).