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 C₀ 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 Calderón-Zygmund operator T on L²(μ) is equivalent to that of T from the localized atomic Hardy space h¹(μ) to L^1,∞(μ) or from h¹(μ) to L¹(μ).

Keywords

Calderón-Zygmund operator / localized atomic Hardy space / nondoubling measure

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg