Paper detail

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, $ω$ a concave function on $(0,\infty)$ of strictly lower type $p_ω\in (0, 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for all $t\in (0,\infty).$ The authors introduce the Orlicz-Hardy space $H_{ω,L}({\mathcal X})$ via the Lusin area function associated to the heat semigroup, and the BMO-type space ${\mathop\mathrm{BMO}_{ρ,L}(\mathcal X)}$. The authors then establish the duality between $H_{ω,L}({\mathcal X})$ and $\mathrm{BMO}_{ρ,L}({\mathcal X})$; as a corollary, the authors obtain the $ρ$-Carleson measure characterization of the space ${\mathop\mathrm{BMO}_{ρ,L}(\mathcal X)}$. Characterizations of $H_{ω,L}({\mathcal X})$, 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=-Δ+V$ be a Schrödinger operator, where $V\in L^1_{\mathrm{\,loc\,}}({\mathbb R}^n)$ is a nonnegative potential. As applications, the authors show that the Riesz transform $\nabla L^{-1/2}$ is bounded from $H_{ω,L}({{\mathbb R}^n})$ to $L(ω)$; moreover, if there exist $q_1,\,q_2\in (0,\infty)$ such that $q_1<1<q_2$ and {\normalsize$[ω(t^{q_2})]^{q_1}$} is a convex function on $(0,\infty)$, then several characterizations of the Orlicz-Hardy space $H_{ω,L}({{\mathbb R}^n})$, 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 $ω(t)=t^p$ for all $t\in (0,\infty)$ and $p\in (0,1)$.