BOUNDEDNESS OF STEIN'S SQUARE FUNCTIONS ASSOCIATED TO OPERATORS ON HARDY SPACES

Let (X, d, mu) be a metric measure space endowed with a metric d and a nonnegative Borel doubling measure A. Let L be a second order non-negative self-adjoint operator on L-2(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 G(delta)(L) arising from Bochner-Riesz means associated to L is bounded from the Hardy spaces H-L(p)(X) to L-P(X) for all 0 <p <= 1.