In this paper the authors establish the boundedness of Calderón-Zygmund operator associated to a non-negative Radon measure without the doubling condition on the Hardy space . More precisely, the Euclidean space is endowed with a non-negative Radon measure which only satisfies the following growth condition that there exists such that
for all and , where , is a fixed number and . For such measure it is not necessary to be doubling. Let be a function on satisfying for ,
and for ,
where and is a constant. The Calderón-Zygmund operator associated to the above kernel and is defined by
For we denote by the truncated operators of . If the operators are bounded on uniformly on , is bounded on . In this case there is an operator
which is the weak limit as of some subsequences of operators . In the main theorem the authors prove that if is bounded on and , then is bounded on . Here, implies that for any bounded function with compact support and , . They adapt the Hardy space as the characterization of a grand maximal function developed by X. Tolsa in [Trans. Am. Math. Soc. 355, No. 1, 315–348 (2003; Zbl 1021.42010)] and their new atomic characterization.