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Musielak-Orlicz Hardy Spaces Associated with Operators and Their Applications

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a nonnegative self-adjoint operator in $L^2(\mathcal{X})$ satisfying the Davies-Gaffney estimates. Let $φ:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that $φ(x,\cdot)$ is an Orlicz function, $φ(\cdot,t)\in A_{\infty}(\mathcal{X})$ (the class of Muckenhoupt weights) and its uniformly critical lower type index $i(φ)\in(0,1]$. In this paper, the authors introduce a Musielak-Orlicz Hardy space $H_{φ,\,L}(\mathcal{X})$ by the Lusin area function associated with the heat semigroup generated by $L$, and a Musielak-Orlicz $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{BMO}_{φ,\,L}(\mathcal{X})$ which is further proved to be the dual space of $H_{φ,\,L}(\mathcal{X})$; as a corollary, the authors obtain the $φ$-Carleson measure characterization of $\mathop\mathrm{BMO}_{φ,\,L}(\mathcal{X})$. Characterizations of $H_{φ,\,L}(\mathcal{X})$, including the atom, the molecule and the Lusin area function associated with the Poisson semigroup of $L$, are presented. Using the atomic characterization, the authors characterize $H_{φ,\,L}(\mathcal{X})$ in terms of $g^\ast_{λ,\,L}$. As further applications, the authors obtain several equivalent characterizations of the Musielak-Orlicz Hardy space $H_{φ,\,L}(\mathbb{R}^n)$ associated with the Schrödinger operator $L=-Δ+V$, where $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ is a nonnegative potential, in terms of the Lusin-area function, the non-tangential maximal function, the radial maximal function, the atom and the molecule.