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

Let \((X, d, \mu )\) $(X,d,\mu )$ be a metric space with a metric d and a doubling measure \(\mu \) $\mu$ . Assume that the operator L generates a bounded holomorphic semigroup \(e^{-tL}\) $e^{-tL}$ on \(L^2(X)\) $L^2(X)$ whose semigroup kernel satisfies the Gaussian upper bound. Also assume that L has a bounded holomorphic functional calculus on \(L^2(X)\) $L^2(X)$ . Then the Hardy spaces \(H^p_L(X)\) $H_L^p(X)$ associated with the operator L can be defined for \(0 < p \le 1\) $0<p\le 1$ . In this paper, we revisit the Calderón-Zygmund decomposition and show that a function \(f \in L^1(X)\cap L^2(X)\) $f\in L^1(X)\cap L^2(X)$ can be decomposed into a good part which is an \(L^{\infty }\) $L^{\infty }$ function and a bad part which is in \(H^p_L(X)\) $H_L^p(X)$ for some \(0< p <1\) $0<p<1$ . An important result of our variants of Calderón-Zygmund decompositions is that if a sub-linear operator T is bounded from \(L^r(X)\) $L^r(X)$ to \(L^r(X)\) $L^r(X)$ for some \(r > 1\) $r>1$ and also bounded from \(H^p_L(X)\) $H_L^p(X)$ to \(L^p(X)\) $L^p(X)$ for some \(0< p < 1\) $0<p<1$ , then T is of weak type (1, 1) and bounded from \(L^q(X)\) $L^q(X)$ to \(L^q(X)\) $L^q(X)$ for all \(1< q <r\) $1<q<r$ .