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Abstract
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 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_{L}^{p}(X) associated with the operator L can be defined for 0<p≤1. In this paper, we revisit the CalderónZygmund decomposition and show that a function f∈L^{1}(X)∩L^{2}(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 CalderónZygmund decompositions is that if a sublinear 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.
Original language  English 

Journal  Potential Analysis 
DOIs  
Publication status  Epub ahead of print  27 Jul 2024 
Keywords
 Hardy spaces
 Weak type (1, 1)
 Heat kernels
Projects
 1 Active

DP22: Harmonic analysis of Laplacians in curved spaces
Li, J., Bui, T., Duong, X., Cowling, M., Ottazzi, A. & Wick, B.
26/04/22 → 25/04/25
Project: Research