<i> L ^p </i> estimates for the bilinear Hilbert transform

Abstract

For the bilinear Hilbert transform given by: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathit{H\hspace{.167em}fg}}({\mathit{x}}){\mathrm{\hspace{.167em}=\hspace{.167em}p.v.}}{\int }{\mathit{f}}({\mathit{x\hspace{.167em}-\hspace{.167em}y}}){\mathit{g}}({\mathit{x\hspace{.167em}+\hspace{.167em}y}}){\mathrm{\hspace{.167em}}}\frac{{\mathit{dy}}}{{\mathit{y}}}{\mathrm{,}}\end{equation*}\end{document} we announce the inequality ∥ H fg ∥ p 3 ≤ K p 1 , p 2 ∥ f ∥ p 1 ∥ g ∥ p 2 , provided 2 &lt; p 1 , p 2 &lt; ∞, 1/ p 3 = 1/ p 1 + 1/ p 2 and 1 &lt; p 3 &lt; 2.

Locations

• Proceedings of the National Academy of Sciences - View
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