Commutators on dyadic martingales. (English) Zbl 0596.47024

For an integrable function f on [0,1) and \(\alpha\in R\) the authors define the fractional integral \(I^{\alpha}f\) which, if \(\alpha >0\), is simply a martingale transform introduced by D. L. Burkholder. The authors show that if \(1<p<q<\infty\) and \(\alpha =1/p-1/q\) then \(b\in BMO\) if and only if the commutator \([b,I^{\alpha}]\) belongs to \(B(L^ p,L^ q)\). This generalizes the corresponding result in Euclidean spaces by S. Chanillo, R. Rochberg and G. Weiss, and Y. Komori. Analogous results on \(H^ p\) martingales and Lipschitz spaces are also given.
Reviewer: H.Tanabe


47B38 Linear operators on function spaces (general)
47B47 Commutators, derivations, elementary operators, etc.
60G46 Martingales and classical analysis
60G42 Martingales with discrete parameter
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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