Musielak-Orlicz Hardy Spaces Associated with Operators and Their Applications

Abstract

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 $\ddot{I}†:\mathcal{X}\times [0,\infty )\to [0,\infty )$ be a function such that $\ddot{I}†(x,\cdot )$ is an Orlicz function, $\ddot{I}†(\cdot ,t)\in A_{\infty }(\mathcal{X})$ (the class of Muckenhoupt weights) and its uniformly critical lower type index $i(\ddot{I}†)\in (0,1]$. In this paper, the authors introduce a Musielak-Orlicz Hardy space $H_{\ddot{I}†,L}(\mathcal{X})$ by the Lusin area function associated with the heat semigroup generated by $L$, and a Musielak-Orlicz $\mathrm{BMO}$-type space ${\mathrm{BMO}}_{\ddot{I}†,L}(\mathcal{X})$ which is further proved to be the dual space of $H_{\ddot{I}†,L}(\mathcal{X})$; as a corollary, the authors obtain the $\ddot{I}†$-Carleson measure characterization of ${\mathrm{BMO}}_{\ddot{I}†,L}(\mathcal{X})$. Characterizations of $H_{\ddot{I}†,L}(\mathcal{X})$, 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 $H_{\ddot{I}†,L}(\mathcal{X})$ in terms of $g_{\hat{I}»,L}^{\ast }$. As further applications, the authors obtain several equivalent characterizations of the Musielak-Orlicz Hardy space $H_{\ddot{I}†,L}({\mathbb{R}}^n)$ associated with the Schrödinger operator $L=-\hat{I}”+V$, where $0\le V\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$ 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.

J. Geom. Anal. (to appear)