An atomic decomposition of the Haj{\l}asz Sobolev space $\Mone$ on manifolds

TL;DR

This paper establishes an atomic decomposition for the Haj{ }asz Sobolev space  on manifolds with doubling measures, linking it to Hardy-Sobolev spaces via atoms under Poincare9 inequality conditions.

Contribution

It provides the first atomic decomposition characterization of  Sobolev spaces on manifolds, extending the theory to non-Euclidean settings with doubling measures.

Findings

Atomic decomposition of  space established

Equivalence between Haj{ }asz  space and Hardy-Sobolev space shown

Decomposition results for both homogeneous and nonhomogeneous spaces

Abstract

Several possible notions of Hardy-Sobolev spaces on a Riemannian manifold with a doubling measure are considered. Under the assumption of a Poincar\'e inequality, the space $\Mone$, defined by Haj{\l}asz, is identified with a Hardy-Sobolev space defined in terms of atoms. Decomposition results are proved for both the homogeneous and the nonhomogeneous spaces.