Calderón-Zygmund decomposition, Hardy spaces associated with operators and weak type estimates

Let (X,d,μ) be a metric space with a metric d and a doubling measure μ. Assume that the operator L generates a bounded holomorphic semigroup e^-tL on L²(X) whose semigroup kernel satisfies the Gaussian upper bound. Also assume that L has a bounded holomorphic functional calculus on L²(X). Then the Hardy spaces H_L^p(X) associated with the operator L can be defined for 0 < p ≤ 1. In this paper, we revisit the Calderón-Zygmund decomposition and show that a function f ∈ L¹(X) ∩ L²(X) can be decomposed into a good part which is an L^∞ function and a bad part which is in H_L^p(X) for some 0 < p < 1. An important result of our variants of Calderón-Zygmund decompositions is that if a sub-linear operator T is bounded from L^r(X) to L^r(X) for some r>1 and also bounded from H_L^p(X) to L^p(X) for some 0 < p < 1, then T is of weak type (1, 1) and bounded from L^q(X) to L^q(X) for all 1 < q < r.