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

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

2010-11-12

arxiv.org/abs/1011.2937v2

Mathematics

Classical Analysis and ODEs

Canad. J. Math. (to appear)

Scientific paper

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 Calder\'on-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 Calder\'on-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.

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