Tent Spaces Associated with Semigroups of Operators

TL;DR

This paper develops a theory of tent spaces and BMO spaces on general measure spaces using semigroups of positive operators, extending classical harmonic analysis concepts beyond Euclidean settings.

Contribution

It introduces a framework for tent spaces and BMO spaces on abstract measure spaces with semigroups, generalizing duality results without geometric assumptions.

Findings

Established a duality between H^1 and BMO spaces in this general setting

Extended results to noncommutative L^p spaces

Derived a BMO-H^1 duality inequality without monotone assumptions

Abstract

We study tent spaces on general measure spaces $(\Omega, \mu)$. We assume that there exists a semigroup of positive operators on $L^p(\Omega, \mu)$ satisfying a monotone property but do not assume any geometric/metric structure on $\Omega$. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman's $H^1$-BMO duality theory. We also get a $H^1$-BMO duality inequality without assuming the monotone property. All the results are proved in a more general setting, namely for noncommutative $L^p$ spaces.