Martingales and the Littlewood-Paley Theorem

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 90)


We have seen that it is possible to prove a quite general Littlewood-Paley theorem (Theorem 4.2.8) for certain disconnected groups by combining the results of Chapter 2 with arguments about topological groups, notably concerning the Bohr compactification. (The Paley theorem on D2 was of course much simpler to establish.) We intend to show now that it i§ possible to adopt an alternative approach, namely to prove a Littlewood-Paley theorem for martingales and then deduce Theorem 4.2.8 from it. Indeed we shall show more: that Theorem 4.2.8 is valid even without the condition of finiteness of the indices of X n in Xn+1, (n ∊ ℤ). This approach has a commendable directness and an elementary character. Moreover, it affords an introduction to the relation between martingales and Littlewood-Paley theory which is only just beginning to be systematically explored. See for instance [39].


Conditional Expectation Weak Type Dual Group Finite Measure Strong Type 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1977

Authors and Affiliations

  1. 1.Institute of Advanced StudiesAustralian National UniversityCanberraAustralia
  2. 2.Flinders UniversityBedford ParkAustralia

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