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Martingale Transforms and Related Singular Integrals

Rodrigo Bañuelos
Transactions of the American Mathematical Society
Vol. 293, No. 2 (Feb., 1986), pp. 547-563
DOI: 10.2307/2000021
Stable URL: http://www.jstor.org/stable/2000021
Page Count: 17
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Martingale Transforms and Related Singular Integrals
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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 Rn. 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 $L^p(R^n), 1 < p < \infty$, 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 $L^p(R^n), 1 < p < \infty$, with constants independent of n.

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