The non-resonant bilinear Hilbert--Carleson operator

2021/06/17 by Cristina Benea, Benea, Cristina, Frederic Bernicot +5
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA

paper · pdf · doi:10.48550/arxiv.2106.09697

144 pages

arxiv created 2021/06/17 · arxiv updated 2021/06/18

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_λ∈ \mathbb R |∫ f(x-t) g(x+t) e^iλta (dt)/(t) | and show that in the non-resonant case a∈ (0,∞)∖\1,2\ the operator BC^a extends continuously from L^p(\mathbb R)× L^q(\mathbb R) into L^r(\mathbb R) whenever (1)/(p)+(1)/(q)=(1)/(r) with 1<p, q≤∞ and (2)/(3)<r<∞. A key novel feature of these operators is that -- in the non-resonant case -- BC^a has a 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 λt^a.

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