Tent Spaces Associated with Semigroups of Operators

We study tent spaces on general measure spaces $(\mathrm{\Omega },\mu )$. We assume that there exists a semigroup of positive operators on $L^p(\mathrm{\Omega },\mu )$ satisfying a monotone property but do not assume any geometric/metric structure on $\mathrm{\Omega }$. 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.