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

Abstract

Abstract Let (𝒳, d , μ) be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that μ(﹛ x ﹜) = 0 for all x ∈ 𝒳. In this paper, we show that the boundedness of a Calderón–Zygmund operator T on L 2 (μ) is equivalent to that of T on L p (μ) for some p ∈ (1,∞), and that of T from L 1 (μ) to L 1,∞ (μ). As an application, we prove that if T is a Calderón–Zygmund operator bounded on L 2 (μ), then its maximal operator is bounded on L p (μ) for all p ∈ (1,∞) and from the space of all complex-valued Borel measures on 𝒳 to L 1,∞ (μ). All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.