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

摘要

Let μ be a nonnegative Radon measure on R~d which satisfies the polynomial growth condition that there exist positive constants C_0 and n ∈ (0, d] such that, for all x ∈ R~d and r ＞ 0, μ(B(x, r)) ≤ C_0r~n, where B(x,r) denotes the open ball centered at x and having radius r. In this paper, we show that, if μ(R~d) ＜ ∞, then the boundedness of a Calderon-Zygmund operator T on L~2 (μ) is equivalent to that of T from the localized atomic Hardy space h~1 (μ) to L~(1,∞)(μ) or from h~1(μ) to L~1(μ).