# Article

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Keywords:
non-negative self-adjoint operator; Stein's square function; Bochner-Riesz means; Davies-Gaffney estimate; molecule Hardy space
Summary:
Let $(X, d, \mu )$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator of order $m$ on $L^2(X)$. Assume that the semigroup ${\rm e}^{-tL}$ 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 ${\mathcal G}_{\delta }(L)$ 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 $0<p\leq 1$.
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