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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 $ω$ 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 $φ:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that $φ(x,\cdot)$ is an Orlicz function, $φ(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index $I(φ)\in(0,1]$ and $φ(\cdot,t)$ satisfies the uniformly reverse Hölder inequality of order $(q_L/I(φ))'$. In this paper, the authors introduce a Musielak-Orlicz-Hardy space $H_{φ,\,L}(\mathcal{X})$, 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 $H_{φ,\,L}(\mathcal{X})$ is also obtained. Furthermore, a sufficient condition for the equivalence between $H_{φ,\,L}(\mathbb{R}^n)$ and the classical Musielak-Orlicz-Hardy space $H_φ(\mathbb{R}^n)$ is given. Moreover, for the Musielak-Orlicz-Hardy space $H_{φ,\,L}(\mathbb{R}^n)$ associated with the second order elliptic operator in divergence form on $\rn$ or the Schrödinger operator $L:=-Δ+V$ 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 $\nabla L^{-1/2}$.