Uniform bounds for the bilinear Hilbert transforms.

摘要

In Chapter 1, we prove that the bilinear Hilbert transforms Haf,g x=p.v. Rfx -tg x+at dtt map Lp1R xLp 2R into Lp(R) uniformly in the real parameter alpha when 2 < p1, p2 < infinity and 1=p1p2p1 +p2<2 .;In Chapter 2, we prove that Halpha map Lp1R xLp 2R into Lp(R) uniformly in the real parameter alpha away from a small neighborhood of -1 when 1 < p1,p2 < 2 and 23=p1p 2p1+p2