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

Abstract

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

arXiv commentCommun. Contemp. Math. (to appear)

References in corpus (1)

• Correction to ``Hardy and BMO spaces associated to divergence form elliptic operators"