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Boundedness of CalderónZygmund Operators on Nonhomogeneous Metric Measure Spaces


Published:20110915
Printed: Aug 2012
Tuomas Hytönen,
Department of Mathematics and Statistics, University of Helsinki, Gustaf Hällströmin Katu 2B, Fi00014 Helsinki, Finland
Suile Liu,
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex systems, Ministry of Education, Beijing 100875, People's Republic of China
Dachun Yang,
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex systems, Ministry of Education, Beijing 100875, People's Republic of China
Dongyong Yang,
School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China
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Abstract
Let $({\mathcal X}, d, \mu)$ be a
separable metric measure space satisfying the known upper
doubling condition, the geometrical doubling condition, and the
nonatomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$.
In this paper, we show that the boundedness of a CalderónZygmund
operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on
$L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$
to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a
CalderónZygmund operator bounded on $L^2(\mu)$,
then its maximal operator is bounded on $L^p(\mu)$
for all $p\in (1, \infty)$ and from
the space of all complexvalued Borel measures on
${\mathcal X}$ to $L^{1,\,\infty}(\mu)$.
All these results generalize the corresponding results of Nazarov et al.
on metric spaces with
measures satisfying the socalled polynomial growth condition.
Keywords: 
upper doubling, geometrical doubling, dominating function, weak type $(1, 1)$ estimate, CalderónZygmund operator, maximal operator
upper doubling, geometrical doubling, dominating function, weak type $(1, 1)$ estimate, CalderónZygmund operator, maximal operator
