Orlicz-Hardy Spaces Associated with Operators Satisfying Davies-Gaffney Estimates

Let $\mathcal{X}$ be a metric space with doubling measure, $L$ a nonnegative self-adjoint operator in $L^2(\mathcal{X})$ satisfying the Davies-Gaffney estimate, $\omega$ a concave function on $(0,\mathrm{\infty })$ of strictly lower type $p_{\omega }\in (0,1]$ and $\rho (t)=t^{-1}\mathrm{/}{\omega }^{-1}(t^{-1})$ for all $t\in (0,\infty).$TheauthorsintroducetheOrlicz-Hardyspace$H_{\omega,L}({\mathcal X})$ via the Lusin area function associated to the heat semigroup, and the BMO-type space ${}_{\rho ,L}(\mathcal{X})$. The authors then establish the duality between $H_{\omega ,L}(\mathcal{X})$ and ${\mathrm{B}\mathrm{M}\mathrm{O}}_{\rho ,L}(\mathcal{X})$; as a corollary, the authors obtain the $\rho$-Carleson measure characterization of the space ${}_{\rho ,L}(\mathcal{X})$. Characterizations of $H_{\omega ,L}(\mathcal{X})$, including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let $\mathcal{X}={\mathbb{R}}^n$ and $L=-\mathrm{\Delta }+V$ be a Schr"odinger operator, where $V\in L^1_{$\mathrm{l}\mathrm{o}\mathrm{c}$}({\mathbb R}^n)$ is a nonnegative potential. As applications, the authors show that the Riesz transform $\mathrm{\nabla }{L}^{-1\mathrm{/}2}$ is bounded from $H_{\omega ,L}({\mathbb{R}}^n)$ to $L(\omega )$; moreover, if there exist $q_1,q_2\in (0,\mathrm{\infty })$ such that $q_1<1<q_2$ and {\normalsize$[\omega (t^{q_2}){]}^{q_1}$} is a convex function on $(0,\mathrm{\infty })$, then several characterizations of the Orlicz-Hardy space $H_{\omega ,L}({\mathbb{R}}^n)$, in terms of the Lusin-area functions, the non-tangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when $\omega (t)=t^p$ for all $t\in (0,\mathrm{\infty })$ and $p\in (0,1)$.