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

Hytönen, Tuomas ; Liu, Suile ; Yang, Dachun ; Yang, Dongyong

Canadian journal of mathematics, Tome 64 (2012) no. 4, pp. 892-923

Résumé

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.

Détail

DOI : 10.4153/CJM-2011-065-2

Mots-clés : 42B20, 42B25, 30L99, upper doubling, geometrical doubling, dominating function, weak type (1, 1) estimate, Calderón–Zygmund operator, maximal operator