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

Abstract

Let (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 is an element of X. In this paper, we show that the boundedness of a Calderon-Zygmund operator T on L-2(mu) is equivalent to that of T on L-p(mu) for some p is an element of (1, infinity), and that of T from L-1(mu) to L-1,L- infinity (mu). As an application, we prove that if T is a Calderon-Zygmund operator bounded on L-2(mu), then its maximal operator is bounded on L-p(mu) for all p is an element of (1, infinity) and from the space of all complex-valued Borel measures on X to L-1,L- infinity (mu). All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.