Uniform bounds for the bilinear Hilbert transforms, I

• 成果类型: Article
• 署名作者: Grafakos, L; Li, XC
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2004.159.889
• 发表日期: 2004
• 页码: 889-933
• 关键词: operators

摘要

It is shown that the bilinear Hilbert transforms H-alpha,H-beta(f, g) (x) = p. v. integral(R) f(x - alphat)g(x - betat) dt/t map L-p1(R) x L-p2(R) --> L-p(R) uniformly in the real parameters alpha,beta when 2 < p(1),p(2) < infinity and 1 < p = p(1)p(2)/p(1)+p(2) < 2. Combining this result with the main result in [9], we deduce that the operators H-1,H-alpha map L-2(R) x L-infinity(R) --> L-2(R) uniformly in the real parameter alpha is an element of [0, 1]. This completes a program initiated by A. Calderon.