$L^p$ estimates for the bilinear Hilbert transform for $1/2<p\le 2/3$: A counterexample and generalizations to non-smooth symbols

Abstract

M. Lacey and C. Thiele proved in [27] (Annals of Math. (1997)) and [28] (Annals of Math. (1999)) that the bilinear Hilbert transform maps $L^{p_1}\times L^{p_2}\to L^p$ boundedly when $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$ with $1<p_1,p_2\le \infty$ and $\frac{2}{3}<p<\infty$. Whether the $L^p$ estimates hold in the range $p\in (1/2,2/3]$ has remained an open problem since then. In this paper, we prove that the bilinear Hilbert transform does not map $\mathcal{F}{L}^{p_1^{\prime }}\times L^{p_2}\to L^p$ for $p_1<2$ and $L^{p_1}\times \mathcal{F}{L}^{p_2^{\prime }}\to L^p$ for $p_2<2$ boundedly (Theorem 1.2). In particular, this shows that the bilinear Hilbert transform neither maps $\mathcal{F}{L}^{p_1^{\prime }}\times L^{p_2}\to L^p$ nor $L^{p_1}\times \mathcal{F}{L}^{p_2^{\prime }}\to L^p$ for $\frac{1}{2}<p<\frac{2}{3}$. Nevertheless, we can establish $L^p$ estimates for the bilinear Fourier multipliers whose symbols are not identical to but arbitrarily close to that of the bilinear Hilbert transform in the full range $p\in (1/2,\infty )$ (Theorem 1.3).

19 pages. This version replaces the arxiv.org/abs/1409.3875v1. There were errors in the original proof of Theorem 1.2 in the first version. The statement of Theorem 1.2 has been modified and changed. Theorem 1.3 and its proof remain unchanged. arXiv admin note: text overlap with arXiv:1403.0624