The non-resonant bilinear Hilbert-Carleson operator - Benea, Cristina; Bernicot, Frédéric; Lie, Victor; Vitturi, Marco (Advances in mathematics (New York. 1965), 2024)
In this paper we introduce the class of bilinear Hilbert-Carleson operators BCaa>0 defined byBCa(f,g)(x):=supλ∈R⁡|∫f(x−t)g(x+t)eiλtadtt| and show that in the non-resonant case a∈(0,∞)∖{1,2} the operator BCa extends continuously from Lp(R)×Lq(R) into Lr(R) whenever 1p+1q=1r with 1<p,q≤∞ and 23<r<∞. A key novel feature of these operators is that – in the non-resonant case – BCa 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 λta. 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 BCa into suitable versions of “dilated” phase-space BHT-like portraits. The resulting decomposition will produce rank-one families of tri-tiles Pmm 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∈Pm into a four-parameter family of tri-tiles S(P) with each of the resulting tri-tiles s∈S(P) 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 TT⁎ 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 BCa. As a consequence of the above, our proof offers a unifying perspective on the distinct methods for treating the zero/non-zero curvature paradigms.