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Martingale transforms and related singular integrals. (English) Zbl 0591.60045
Summary: The operators obtained by taking conditional expectation of continuous time martingale transforms are studied, both on the circle T and on \({\mathbb{R}}^ n\). Using a Burkholder-Gundy inequality for vector-valued martingales, it is shown that the vector formed by any number of these operators is bounded on \(L^ p({\mathbb{R}}^ n)\), \(1<p<\infty,\) with constants that depend only on p and the norms of the matrices involved. As a corollary we obtain a recent result of E. M. Stein [Bull. Am. Math. Soc., New Ser. 9, 71-73 (1983; Zbl 0515.42018)] on the boundedness of the Riesz transforms on \(L^ p({\mathbb{R}}^ n)\), \(1<p<\infty,\) with constants independent of n.

MSC:
60G44 Martingales with continuous parameter
60H05 Stochastic integrals
60G46 Martingales and classical analysis
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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