Copyright © 2009 Qingying Xue and Juyang Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let μ be a positive Radon measure on ℝd which may be nondoubling. The only condition that μ satisfies is μ(B(x,r))≤C0rn for all x∈ℝd, r>0, and some fixed constant C0. In this paper, we introduce the operator gλ,μ∗ related to such a measure and assume it is bounded on L2(μ). We then establish its boundedness, respectively, from the Lebesgue space L1(μ) to the weak Lebesgue space L1,∞(μ), from the Hardy space H1(μ) to L1(μ) and from the Lesesgue space L∞(μ) to the space RBLO(μ). As a corollary, we obtain the boundedness of gλ,μ∗ in the Lebesgue space Lp(μ) with p∈(1,∞).