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)) ⩽ C₀ 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 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 / non-doubling measure

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