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

  Published:2011-09-15
 Printed: Aug 2012
  • Tuomas Hytönen,
    Department of Mathematics and Statistics, University of Helsinki, Gustaf Hällströmin Katu 2B, Fi-00014 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 non-atomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a Calderón-Zygmund 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 Calderón-Zygmund 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 complex-valued 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 so-called polynomial growth condition.
Keywords: upper doubling, geometrical doubling, dominating function, weak type $(1, 1)$ estimate, Calderón-Zygmund operator, maximal operator upper doubling, geometrical doubling, dominating function, weak type $(1, 1)$ estimate, Calderón-Zygmund operator, maximal operator
MSC Classifications: 42B20, 42B25, 30L99 show english descriptions Singular and oscillatory integrals (Calderon-Zygmund, etc.)
Maximal functions, Littlewood-Paley theory
None of the above, but in this section
42B20 - Singular and oscillatory integrals (Calderon-Zygmund, etc.)
42B25 - Maximal functions, Littlewood-Paley theory
30L99 - None of the above, but in this section
 

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