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

Abstract

Let $(\cx,\,d,\,μ)$ 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 Calderón-Zygmund operator associated with a singular integral whose kernel satisfies the standard size condition and the Hörmander condition, its $L^p(μ)$ boundedness with $p\in(1,\infty)$ is equivalent to its boundedness from $L^1(μ)$ into $L^{1,\infty}(μ)$. Moreover, applying this, together with a new Cotlar type inequality, the authors show that if the Calderón-Zygmund operator $T$ is bounded on $L^2(μ)$, then the corresponding maximal Calderón-Zygmund is bounded on $L^p(μ)$ for all $p\in(1,\infty)$, and bounded from $L^1(μ)$ into $L^{1,\infty}(μ)$. These results essentially improve the existing results.