The full range of uniform bounds for the bilinear Hilbert transform

Abstract

We prove uniform uniform $L^p$ bounds for the family of bilinear Hilbert transforms ${\mathrm{BHT}}_{\beta }[f_1,f_2](x):=p.v.{\int }_{\mathbb{R}}{f}_1(x-t)f_2(x+\beta t)\frac{\mathrm{d}t}{t}$. We show that the operator ${\mathrm{BHT}}_{\beta }$ maps $L^{p_1}\times L^{p_2}$ into $L^p$ as long as $p_1\in (1,\infty )$, $p_2\in (1,\infty )$, and $p>\frac{2}{3}$ with a bound independent of $\beta \in (0,1]$. This is the full open range of exponents where the modulation invariant class of bilinear operators containing ${\mathrm{BHT}}_{\beta }$ can be bounded uniformly. This is done by proving boundedness of certain affine transformations of the frequency-time-scale space ${\mathbb{R}}_{+}^3$ in terms of iterated outer Lebesgue spaces. This results in new linear and bilinear wave packet embedding bounds well suited to study uniform bounds.

107 pages, 1 figure. The authors welcome comments, corrections, and questions