The critical line for modulation invariant bilinear multipliers

Abstract

We prove that the bilinear multipliers with symbols singular along a nondegenerate line, in the class of Muscalu, Tao and Thiele, map $L^{p_1}(\mathbb R)\times L^{p_2}(\mathbb R)$ into $L^{2/3,\infty}(\mathbb R)$ on the whole critical line \[ \frac1{p_1}+\frac1{p_2}=\frac32,\qquad 1<p_1,p_2<2 . \] This is sharp, as no estimate into $L^{p,\infty}$ with $p<\frac23$ holds uniformly over the class. Moreover, the generalized restricted weak type estimates of the class extend, without logarithmic loss, to the open short sides of the hexagon of exponents. For the bilinear Hilbert transforms, this confirms a folklore conjecture recorded by Thiele and the author, and removes the doubly logarithmic term in their endpoint estimates. The main step is an improvement over the multi-frequency Calderón--Zygmund decompositions already present in the literature. The bad part of this version has vanishing moments of order one at the top frequencies of the trees close to each Whitney interval, and vanishing moments of logarithmic order at the frequency bins where the tops of distant trees accumulate.