Calderón-Zygmund Decomposition, Hardy Spaces Associated with Operators and Weak Type Estimates

Let \((X, d, \mu )\) be a metric space with a metric d and a doubling measure \(\mu \). Assume that the operator L generates a bounded holomorphic semigroup \(e^{-tL}\) on \(L^2(X)\) whose semigroup kernel satisfies the Gaussian upper bound. Also assume that L has a bounded holomorphic functional calculus on \(L^2(X)\). Then the Hardy spaces \(H^p_L(X)\) associated with the operator L can be defined for \(0 < p \le 1\). In this paper, we revisit the Calderón-Zygmund decomposition and show that a function \(f \in L^1(X)\cap L^2(X)\) can be decomposed into a good part which is an \(L^{\infty }\) function and a bad part which is in \(H^p_L(X)\) 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 \(H^p_L(X)\) 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 \(1< q <r\).