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

Let µ be a nonnegative Radon measure on ℝd which satisfies the polynomial growth condition that there exist positive constants C0 and n ∈ (0, d] such that, for all x ∈ ℝd and r > 0, µ(B(x, r)) ⩽ C0rn, where B(x, r) denotes the open ball centered at x and having radius r. In this paper, we show that, if µ(ℝd) < ∞, then the boundedness of a Calderón-Zygmund operator T on L2(µ) is equivalent to that of T from the localized atomic Hardy space h1(µ) to L1,∞(µ) or from h1(µ) to L1(µ).