Martingale transforms and related singular integrals

Bañuelos, Rodrigo

doi:10.1090/S0002-9947-1986-0816309-0

Martingale transforms and related singular integrals
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by Rodrigo Bañuelos

Trans. Amer. Math. Soc. 293 (1986), 547-563

DOI: https://doi.org/10.1090/S0002-9947-1986-0816309-0

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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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