A note on local Hardy spaces

TL;DR

This paper explores the relationship between local and global Hardy spaces associated with a specific class of operators on metric measure spaces, showing they coincide under certain spectral conditions.

Contribution

It demonstrates that local and global Hardy spaces are equivalent when the operator has a spectral gap, extending understanding of Hardy spaces in this context.

Findings

Local and global Hardy spaces coincide under spectral gap conditions

Operators with generalized Gaussian estimates are studied in doubling metric spaces

Spectral gap implies equivalence of Hardy space definitions

Abstract

We consider a non-negative self-adjoint operator L satisfying generalized Gaussian estimates on a doubling metric measure space, and show that if L has a spectral gap then the local and global Hardy spaces defined by means of appropriate square functions coincide.