Resumen de Atomic and maximal function characterizations of Musielak–Orlicz–Hardy spaces associated to non-negative self-adjoint operators on spaces of homogeneous type
Sibei Yang, Dachun Yang
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.
