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

Abstract

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 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 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).