Tent spaces associated with semigroups of operators

We study tent spaces on general measure spaces (Ω,μ). We assume that there exists a semigroup of positive operators on Lp(Ω,μ) satisfying a monotone property but do not assume any geometric/metric structure on Ω. 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 H¹-BMO duality theory. We also get a H¹-BMO duality inequality without assuming the monotone property. All the results are proved in a more general setting, namely for noncommutative Lp spaces.