Endpoint estimates for homogeneous Littlewood-Paleyg-functions with non-doubling measures

Let µ be a nonnegative Radon measure on ℝd which satisfies thegrowth condition that there exist constants C0 > 0 and n ∈ (0, d] such thatfor all x ∈ ℝd and r > 0, μ(B(x,r))≤C0rn, where B(x, r) is the open ballcentered at x and having radius r . In this paper, when ℝd is not an initial cubewhich implies µ(ℝd) = ∞, the authors prove that the homogeneous Littlewood-Paley g-function of Tolsa is bounded from the Hardy space H1 (µ) to L1(µ), andfurthermore, that if f ∈ RBMO (µ), then [ġ(f )]2 is either infinite everywhereor 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