Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates

Abstract

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a one-to-one operator of type $\ddot{I}‰$ 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 $\ddot{I}†:\mathcal{X}\times [0,\infty )\to [0,\infty )$ be a function such that $\ddot{I}†(x,\cdot )$ is an Orlicz function, $\ddot{I}†(\cdot ,t)\in {\mathbb{A}}_{\infty }(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index $I(\ddot{I}†)\in (0,1]$ and $\ddot{I}†(\cdot ,t)$ satisfies the uniformly reverse Hölder inequality of order $(q_L/I(\ddot{I}†){)}^{\prime }$. In this paper, the authors introduce a Musielak-Orlicz-Hardy space $H_{\ddot{I}†,L}(\mathcal{X})$, 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 $H_{\ddot{I}†,L}(\mathcal{X})$ is also obtained. Furthermore, a sufficient condition for the equivalence between $H_{\ddot{I}†,L}({\mathbb{R}}^n)$ and the classical Musielak-Orlicz-Hardy space $H_{\ddot{I}}†({\mathbb{R}}^n)$ is given. Moreover, for the Musielak-Orlicz-Hardy space $H_{\ddot{I}†,L}({\mathbb{R}}^n)$ associated with the second order elliptic operator in divergence form on $\rn$ or the Schrödinger operator $L:=-\hat{I}”+V$ with $0\le V\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$, 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 $\nabla L^{-1/2}$.

Published in Analysis and Geometry in Metric Spaces, volume 1 (2012), 69-129. arXiv admin note: text overlap with arXiv:1201.5512