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

Abstract

AbstractLet $$(X, d, \mu )$$ ( X , d , μ ) be a metric space with a metric d and a doubling measure $$\mu $$ μ . Assume that the operator L generates a bounded holomorphic semigroup $$e^{-tL}$$ e - t L 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 ≤ 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 ∈ L 1 ( X ) ∩ L 2 ( X ) can be decomposed into a good part which is an $$L^{\infty }$$ L ∞ 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 .