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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 YangEmail author
Article
  • 13 Downloads

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.

Keywords

Musielak–Orlicz–Hardy space Atom Maximal function Non-negative self-adjoint operator Gaussian upper bound estimate Space of homogeneous type Strongly Lipschitz domain 

Mathematics Subject Classification

Primary 42B25 Secondary 42B35 46E30 30L99 

Notes

Acknowledgements

The authors would like to thank the referee for her/his very careful reading and several valuable comments which indeed improve the presentation of this article.

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Authors and Affiliations

  1. 1.School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex SystemsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China

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