Boundedness of Maximal Calder\'on-Zygmund Operators on Non-homogeneous Metric Measure Spaces

Let $(\cx,\,d,\,\mu)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors show that for the maximal Calder\'on-Zygmund operator associated with a singular integral whose kernel satisfies the standard size condition and the H\"ormander condition, its $L^p(\mu)$ boundedness with $p\in(1,\infty)$ is equivalent to its boundedness from $L^1(\mu)$ into $L^{1,\infty}(\mu)$. Moreover, applying this, together with a new Cotlar type inequality, the authors show that if the Calder\'on-Zygmund operator $T$ is bounded on $L^2(\mu)$, then the corresponding maximal Calder\'on-Zygmund is bounded on $L^p(\mu)$ for all $p\in(1,\infty)$, and bounded from $L^1(\mu)$ into $L^{1,\infty}(\mu)$. These results essentially improve the existing results.