The non-resonant bilinear Hilbert--Carleson operator

TL;DR

This paper introduces a class of bilinear Hilbert--Carleson operators and proves their boundedness in certain non-resonant cases, revealing a hybrid nature combining features of classical bilinear Hilbert transforms and nonlinear phase operators.

Contribution

The paper defines the bilinear Hilbert--Carleson operators $BC^a$ and establishes their boundedness for non-resonant exponents, demonstrating a novel hybrid curvature property.

Findings

Operators are bounded on $L^p \times L^q \to L^r$ for specified ranges.

Non-resonant case excludes $a=1,2$, ensuring boundedness.

Operators exhibit combined zero and non-zero curvature features.

Abstract

In this paper we introduce the class of bilinear Hilbert--Carleson operators $\{BC^a\}_{a>0}$ defined by $$ BC^{a}(f,g)(x):= \sup_{\lambda\in {\mathbb R}} \Big|\int f(x-t)\, g(x+t)\, e^{i\lambda t^a} \, \frac{dt}{t} \Big| $$ and show that in the non-resonant case $a\in (0,\infty)\setminus\{1,2\}$ the operator $BC^a$ extends continuously from $L^p({\mathbb R})\times L^q({\mathbb R})$ into $L^r({\mathbb R})$ whenever $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ with $1<p,\,q\leq\infty$ and $\frac{2}{3}<r<\infty$. A key novel feature of these operators is that -- in the non-resonant case -- $BC^{a}$ has a \emph{hybrid} nature enjoying both (1) ``zero curvature'' features inherited from the modulation invariance property of the classical bilinear Hilbert transform (BHT), and (2) ``non-zero curvature'' features arising from the Carleson-type operator with nonlinear phase $\lambda t^a$.