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Boundedness of Calderón–Zygmund Operators on Non-homogeneous Metric Measure Spaces

Published online by Cambridge University Press:  20 November 2018

Tuomas Hytönen ,
Suile Liu ,
Dachun Yang  and
Dongyong Yang
Tuomas Hytönen
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, Gustaf H¨allstr¨omin Katu 2B, Fi-00014 Helsinki, Finland email: tuomas.hytonen@helsinki.fi
Suile Liu
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex systems, Ministry of Education, Beijing 100875, People’s Republic of China email: slliu@mail.bnu.edu.cndcyang@bnu.edu.cn
Dachun Yang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex systems, Ministry of Education, Beijing 100875, People’s Republic of China email: slliu@mail.bnu.edu.cndcyang@bnu.edu.cn
Dongyong Yang
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, People’s Republic of China email: dyyang@xmu.edu.cn
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Abstract

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Let $\left( \text{ }\!\!\chi\!\!\text{ ,}\,d,\,\mu \right)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that $\mu \left( \left\{ x \right\} \right)\,=\,0$ for all $x\,\in \,\text{ }\!\!\chi\!\!\text{ }$. In this paper, we show that the boundedness of a Calderón–Zygmund operator $T$ on ${{L}^{2}}\left( \mu \right)$ is equivalent to that of $T$ on ${{L}^{p}}\left( \mu \right)$ for some $p\,\in \,\left( 1,\,\infty \right)$, and that of $T$ from ${{L}^{1}}\left( \mu \right)$ to ${{L}^{1,\,\infty }}\left( \mu \right)$. As an application, we prove that if $T$ is a Calderón–Zygmund operator bounded on ${{L}^{2}}\left( \mu \right)$, then its maximal operator is bounded on ${{L}^{p}}\left( \mu \right)$ for all $p\,\in \,\left( 1,\,\infty \right)$ and from the space of all complex-valued Borel measures on $\text{ }\!\!\chi\!\!\text{ }$ to ${{L}^{1,\,\infty }}\left( \mu \right)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

Keywords

42B2042B2530L99upper doublinggeometrical doublingdominating functionweak type (1, 1) estimateCalderón–Zygmund operatormaximal operator
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 4 , 01 August 2012 , pp. 892 - 923
Copyright
Copyright © Canadian Mathematical Society 2012

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