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

By Renjin Jiang, Dachun Yang. 2009

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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 ${\mathop\mathrm{BMO}_{\rho,L}(\mathcal X)}$. 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 ${\mathop\mathrm{BMO}_{\rho,L}(\mathcal X)}$. 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 Schr\"odinger 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 $H_{\omega,L}({{\mathbb R}^n})$ 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 $p\in (0,1)$.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6b129969edae1ada208e4298c7a844aca203fb28

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In Applied Mathematics

  1. Dachun Yang
  2. Sibei Yang
  3. Yiyu Liang
  4. Luong Dang Ky
  5. Ji Li
  6. Xuan Thinh Duong
  7. The Anh Bui
  8. Wen Yuan

In Mathematical Physics

  1. Dachun Yang
  2. Sibei Yang
  3. Luong Dang Ky
  4. Ji Li
  5. Xuan Thinh Duong
  6. The Anh Bui
  7. Yiyu Liang
  8. Wen Yuan

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