Hardy spaces and BMO-functions induced by ergodic-flows (Q1820363)

The author studies the Hardy spaces \(H^ p(X)\) associated with ergodic flows which were introduced by \textit{R. R. Coifman} and \textit{G. Weiss} [Proc. Nat. Acad. Sci. USA 70, 1761-1763 (1973; Zbl 0257.46077)]. The main attention lies on the ergodic generalization of BMO functions. In particular, a new proof is given of the fact that the dual of \(H^ 1(X)\) is BMO(X). Some unusual properties of the ergodic Hilbert transform are discussed. The author also extends a theorem of \textit{K. Petersen} [Proc. Am. Math. Soc. 79, 549-555 (1980; Zbl 0442.28020)] on the existence of unbounded (ergodic) VMO functions. The last section of the paper contains another proof of the result that \(H^{\infty}(X)\) is a weak-*-Dirichlet algebra.