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Martingale operators and Hardy spaces generated by them. (English) Zbl 0822.60043

Summary: Martingale Hardy spaces and BMO spaces generated by an operator \(T\) are investigated. An atomic decomposition of the space \(H^ T_ p\) is given if the operator \(T\) is predictable. We generalize the John-Nirenberg theorem, namely, we prove that the \(\text{BMO}_ q\) spaces generated by an operator \(T\) are all equivalent. The sharp operator is also considered and it is verified that the \(L_ p\) norm of the sharp operator is equivalent to the \(H^ T_ p\) norm. The interpolation spaces between the Hardy and BMO spaces are identified by the real method. Martingale inequalities between Hardy spaces generated by two different operators are considered. In particular, we obtain inequalities for the maximal function, for the \(q\)-variation and for the conditional \(q\)-variation. The duals of the Hardy spaces generated by these special operators are characterized.

MSC:

60G42 Martingales with discrete parameter
46B70 Interpolation between normed linear spaces
42B30 \(H^p\)-spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
60G46 Martingales and classical analysis
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