Atomic and maximal function characterizations of Musielak–Orlicz–Hardy spaces associated to non-negative self-adjoint operators on spaces of homogeneous type

Abstract

Let ${\mathcal {X}}$ be a metric space with doubling measure and L be a non-negative self-adjoint operator on $L^2({\mathcal {X}})$ whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function $\varphi :\ {\mathcal {X}}\times [0,\infty ) \rightarrow [0,\infty )$ satisfies that $\varphi (x,\cdot )$ is an Orlicz function and $\varphi (\cdot ,t)\in {{\mathbb {A}}}_{\infty }({\mathcal {X}})$ (the class of uniformly Muckenhoupt weights). Let $H_{\varphi ,\,L}({\mathcal {X}})$ be the Musielak–Orlicz–Hardy space defined via the Lusin area function associated with the heat semigroup of L. In this article, the authors characterize the space $H_{\varphi ,\,L}({\mathcal {X}})$ by means of atoms, non-tangential and radial maximal functions associated with L. In particular, when $\mu ({\mathcal {X}})<\infty$, the local non-tangential and radial maximal function characterizations of $H_{\varphi ,\,L}({\mathcal {X}})$ are obtained. As applications, the authors obtain various maximal function and the atomic characterizations of the “geometric” Musielak–Orlicz–Hardy spaces $H_{\varphi ,\,r}(\Omega )$ and $H_{\varphi ,\,z}(\Omega )$ on the strongly Lipschitz domain $\Omega$ in ${\mathbb {R}}^n$ associated with second-order self-adjoint elliptic operators with the Dirichlet and the Neumann boundary conditions; even when $\varphi (x,t):=t$ for any $x\in {\mathbb {R}}^n$ and $t\in [0,\infty )$, the equivalent characterizations of $H_{\varphi ,\,z}(\Omega )$ given in this article improve the known results via removing the assumption that $\Omega$ is unbounded.