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Lp estimates on the bilinear Hilbert transform for 2 < p < \infty
In this paper we prove a conjecture of A. Calderon that the bilinear Hilbert transform can be estimated using Lp norms.The bilinear Hilbert transform of two functions $f$ and $g$ on the real line is defined as $H(f,g)(x):=p.v. \int f(x-t)g(x+t) t^{-1} dt$. If $2 < p,q < /infty$, $1 < r < 2$, and $p^(-1)+q^(-1)=r^(-1)$, then the $L^r$ norm of $H(f,g)$ is bounded by a constant times the product of the Lp norm of $f$ and the $L^q$ norm of $g$. In this statement the range of exponents is not meant to be sharp.
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