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

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a one-to-one operator of type $\omega$ having a bounded $H_{\mathrm{\infty }}$-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 $q_L\in (2,\mathrm{\infty }]$. Let $\phi :\mathcal{X}\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be a function such that $\phi (x,\cdot )$ 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 $I(\phi )\in (0,1]$ and $\phi (\cdot ,t)$ satisfies the uniformly reverse H"older inequality of order $(q_L\mathrm{/}I(\phi ){)}^{\mathrm{\prime }}$. In this paper, the authors introduce a Musielak-Orlicz-Hardy space $H_{\phi ,L}(\mathcal{X})$, 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 $H_{\phi ,L}(\mathcal{X})$ is also obtained. Furthermore, a sufficient condition for the equivalence between $H_{\phi ,L}({\mathbb{R}}^n)$ and the classical Musielak-Orlicz-Hardy space $H_{\phi }({\mathbb{R}}^n)$ is given. Moreover, for the Musielak-Orlicz-Hardy space $H_{\phi ,L}({\mathbb{R}}^n)$ associated with the second order elliptic operator in divergence form on $\text{rn}$ or the Schr"odinger operator $L:=-\mathrm{\Delta }+V$ with $0\le V\in L^1{$\mathrm{l}\mathrm{o}\mathrm{c}$}($\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 $\mathrm{\nabla }{L}^{-1\mathrm{/}2}$.