Boundedness of Caldern-Zygmund operators with finite non-doubling measures

Let A mu be a nonnegative Radon measure on a"e (d) which satisfies the polynomial growth condition that there exist positive constants C (0) and n a (0, d] such that, for all x a a"e (d) and r > 0, A mu(B(x, r)) a (c) 1/2 C (0) r (n) , where B(x, r) denotes the open ball centered at x and having radius r. In this paper, we show that, if A mu(a"e (d) ) < a, then the boundedness of a Caldern-Zygmund operator T on L (2)(A mu) is equivalent to that of T from the localized atomic Hardy space h (1)(A mu) to L (1,a)(A mu) or from h (1)(A mu) to L (1)(A mu).