The non-resonant bilinear Hilbert--Carleson operator

Abstract

In this paper we introduce the class of bilinear Hilbert--Carleson operators $\{B{C}^a{\}}_{a>0}$ defined by $$B{C}^a(f,g)(x):=\underset{\lambda \in \mathbb{R}}{\sup }|\int f(x-t)g(x+t)e^{i\lambda t^a}\frac{dt}{t}|$$ and show that in the non-resonant case $a\in (0,\infty )\setminus \{1,2\}$ the operator $B{C}^a$ 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 $1<p,q\le \infty$ and $\frac{2}{3}<r<\infty$. A key novel feature of these operators is that -- in the non-resonant case -- $B{C}^a$ 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 $\lambda t^a$.

arXiv comment144 pages

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