Boundedness of Stein’s square functions and Bochner-Riesz means associated to operators on hardy spaces

Abstract

Let (X, d, µ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure µ. Let L be a non-negative self-adjoint operator of order m on L 2(X). Assume that the semigroup e−tL generated by L satisfies the Davies-Gaffney estimate of order m and L satisfies the Plancherel type estimate. Let H p L (X) be the Hardy space associated with L. We show the boundedness of Stein’s square function \({g_\delta }(L)\) arising from Bochner-Riesz means associated to L from Hardy spaces H p L (X) to L p(X), and also study the boundedness of Bochner-Riesz means on Hardy spaces H p L (X) for 0 < p ⩽ 1.