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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Martingale transforms and related singular integrals


Author: Rodrigo Bañuelos
Journal: Trans. Amer. Math. Soc. 293 (1986), 547-563
MSC: Primary 60G44; Secondary 42B20, 60G46, 60H05
MathSciNet review: 816309
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Abstract: The operators obtained by taking conditional expectation of continuous time martingale transforms are studied, both on the circle $ T$ and on $ {{\mathbf{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}({{\mathbf{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 Stein on the boundedness of the Riesz transforms on $ {L^p}({{\mathbf{R}}^n}),\,1 < p < \infty $, with constants independent of $ n$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0816309-0
PII: S 0002-9947(1986)0816309-0
Article copyright: © Copyright 1986 American Mathematical Society