Martingales and the Littlewood-Paley Theorem
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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 .
KeywordsConditional Expectation Weak Type Dual Group Finite Measure Strong Type
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