Endpoint estimates for homogeneous Littlewood-Paley g-functions with non-doubling measures

Abstract

Let µ be a nonnegative Radon measure on ℝ^d which satisfies the growth condition that there exist constants C₀ > 0 and n ∈ (0, d] such that for all x ∈ ℝ^d and r > 0, μ(B(x,r))≤C0rn, where B(x, r) is the open ball centered at x and having radius r . In this paper, when ℝ^d is not an initial cube which implies µ(ℝ^d) = ∞, the authors prove that the homogeneous Littlewood-Paley g-function of Tolsa is bounded from the Hardy space H¹ (µ) to L¹(µ), and furthermore, that if f ∈ RBMO (µ), then [ġ(f )]² is either infinite everywhere or finite almost everywhere, and in the latter case, [ġ(f)]² belongs to RBLO (µ) with norm no more than C‖f‖RBMO(μ)2, where C≻0 is independent of f .