Boundedness of Calderón–Zygmund Operators on Non-homogeneous Metric Measure Spaces
Boundedness of Calderón–Zygmund Operators on Non-homogeneous Metric Measure Spaces
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DOI：
10.4153/cjm-2011-065-2
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发表时间：
2010-11
期刊：
影响因子：
--
通讯作者：
T. Hytönen;Suile Liu;Dachun Yang;Dongyong Yang
中科院分区：
文献类型：
--
作者：
T. Hytönen;Suile Liu;Dachun Yang;Dongyong Yang
Abstract 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.
Abstract 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.