Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates
| Autor: | Bui, The Anh, Cao, Jun, Ky, Luong Dang, Yang, Dachun, Yang, Sibei |
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| Rok vydání: | 2013 |
| Předmět: | |
| Zdroj: | Analysis and Geometry in Metric Spaces, volume 1 (2012), 69-129 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2478/agms-2012-0006 |
| Popis: | Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a one-to-one operator of type $\omega$ having a bounded $H_\infty$-functional calculus in $L^2(\mathcal{X})$ satisfying the reinforced $(p_L, q_L)$ off-diagonal estimates on balls, where $p_L\in[1,2)$ and $q_L\in(2,\infty]$. Let $\varphi:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that $\varphi(x,\cdot)$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index $I(\varphi)\in(0,1]$ and $\varphi(\cdot,t)$ satisfies the uniformly reverse H\"older inequality of order $(q_L/I(\varphi))'$. In this paper, the authors introduce a Musielak-Orlicz-Hardy space $H_{\varphi,\,L}(\mathcal{X})$, via the Lusin-area function associated with $L$, and establish its molecular characterization. In particular, when $L$ is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of $H_{\varphi,\,L}(\mathcal{X})$ is also obtained. Furthermore, a sufficient condition for the equivalence between $H_{\varphi,\,L}(\mathbb{R}^n)$ and the classical Musielak-Orlicz-Hardy space $H_{\varphi}(\mathbb{R}^n)$ is given. Moreover, for the Musielak-Orlicz-Hardy space $H_{\varphi,\,L}(\mathbb{R}^n)$ associated with the second order elliptic operator in divergence form on $\rn$ or the Schr\"odinger operator $L:=-\Delta+V$ with $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$, the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors discuss the boundedness of the Riesz transform $\nabla L^{-1/2}$. Comment: Published in Analysis and Geometry in Metric Spaces, volume 1 (2012), 69-129. arXiv admin note: text overlap with arXiv:1201.5512 |
| Databáze: | arXiv |
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