Boundedness of stein's square functions associated to operators on hardy spaces

Let (X, d, μ) be a metric measure space endowed with a metric d and a nonnegative Borel doubling measure μ. Let L be a second order non-negative self-adjoint operator on L2(X). Assume that the semigroup e-tL generated by L satisfies the Davies-Gaffney estimates. Also, assume that L satisfies Plancherel type estimate. Under these conditions, we show that Stein's square function Gδ(L)Gδ(L) arising from Bochner-Riesz means associated to L is bounded from the Hardy spaces Hp(X) to Lp(X) for all 0 < p ≤ 1.