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 $\mathrm{BHT}_{\beta} [f_1, f_2] (x) := \mathrm{p.v.} \int_{\mathbb{R}} f_1 (x - t) f_2 (x + \beta t) \frac{\mathrm{d} t}{t}$. 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.