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

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.

Canad. J. Math. (to appear)