The non-resonant bilinear Hilbert--Carleson operator

Abstract

In this paper we introduce the class of bilinear Hilbert-Carleson operators {BC a } a>0 defined byiλt a dt t Date: June 18, 2021. Key words and phrases. Bilinear Hilbert transform, Carleson operator, wave-packet analysis, time-frequency analysis, zero/non-zero curvature, almost orthogonality, Gabor frame decomposition, phase linearization, time-frequency correlations.42 4. Two-resolution analysis: Discretization 47 5. The high resolution, single-scale analysis (I): Extracting the cancellation encoded in the phase curvature 68 6. The high resolution, single-scale analysis (II): Time-frequency localization via weight mass and input size distribution 94 7. The low resolution, multi-scale analysis: modulation invariance analysis 108 Appendix A. Lack of m-decaying absolute summability for the discrete phase-linearized wave-packet model 128 Appendix B. Analysis of the multiplier 132 References 140 10 If k = 1 one also assumes ϕ ′ monotone. 11 This may serve as a linear prototype for the formulation of Problem A above.12 The presence of the L 2 norm in the right-hand side of (8) as opposed to the L ∞ norm appearing in ( 7) is not of key relevance since, via standard interpolation techniques, one can always establish a correspondence between these two situations.13 It is immediate that if P is degenerate in the sense of ( 9), then the terms in this decomposition can be distributed to the input functions, thus removing any possibility of decay in (7).