The non-resonant bilinear Hilbert-Carleson operator

Abstract

In this paper we introduce the class of bilinear Hilbert-Carleson operators ${\left\{B{C}^a\right\}}_{a>0}$ defined by$B{C}^a(f,g)\left(x\right):=\underset{\lambda \in R}{\sup }|\int f(x-t\left)g\right(x+t\left)e^{i\lambda t^a}\frac{dt}{t}\right|$ and show that in the non-resonant case $a\in (0,\infty )\setminus \{1,2\}$ the operator $B{C}^a$ extends continuously from $L^p\left(R\right)\times L^q\left(R\right)$ into $L^r\left(R\right)$ 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 hybrid nature enjoying both

• (I)

  zero curvature features inherited from the modulation invariance property of the classical bilinear Hilbert transform (BHT), and

• (II)

  non-zero curvature features arising from the Carleson-type operator with nonlinear phase $\lambda t^a$.

In order to simultaneously control these two competing facets of our operator we develop a two-resolution approach:

• •

  A low resolution, multi-scale analysis addressing (I) and relying on the time-frequency discretization of $B{C}^a$ into suitable versions of “dilated” phase-space BHT-like portraits. The resulting decomposition will produce rank-one families of tri-tiles ${\left\{P_m\right\}}_m$ such that the components of any such tri-tile will no longer have area one Heisenberg localization. The control over these families will be obtained via a refinement of the time-frequency methods introduced in [35] and [36].

• •

  A high resolution, single scale analysis addressing (II) and relying on a further discretization of each of the tri-tiles $P\in P_m$ into a four-parameter family of tri-tiles $S\left(P\right)$ with each of the resulting tri-tiles $s\in S\left(P\right)$ now obeying the area one Heisenberg localization. The design of these latter families as well as the extraction of the cancellation encoded in the non-zero curvature of the multiplier's phase within each given P relies on the LGC-methodology introduced in [41].

A further interesting aspect of our work is that the high resolution analysis itself involves two types of decompositions capturing the local (single scale) behavior of our operator:

• •

  A continuous phase-linearized spatial model that serves as the vehicle for extracting the cancellation from the multiplier's phase. The latter is achieved via $T{T}^{⁎}$ arguments, number-theoretic tools (Weyl sums) and phase level set analysis exploiting time-frequency correlations.

• •

  A discrete phase-linearized wave-packet model that takes the just-captured phase cancellation and feeds it into the low resolution analysis in order to achieve the global control over $B{C}^a$.

As a consequence of the above, our proof offers a unifying perspective on the distinct methods for treating the zero/non-zero curvature paradigms.