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 $L^p(\hat{I}¼)$ boundedness with $p\in (1,\infty )$ is equivalent to its boundedness from $L^1(\hat{I}¼)$ into $L^{1,\infty }(\hat{I}¼)$. Moreover, applying this, together with a new Cotlar type inequality, the authors show that if the Calderón-Zygmund operator $T$ is bounded on $L^2(\hat{I}¼)$, then the corresponding maximal Calderón-Zygmund is bounded on $L^p(\hat{I}¼)$ for all $p\in (1,\infty )$, and bounded from $L^1(\hat{I}¼)$ into $L^{1,\infty }(\hat{I}¼)$. These results essentially improve the existing results.

Proc. Roy. Soc. Edinburgh Sect. A. (to appear), 22 pages