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Non-zero to zero curvature transition: Operators along hybrid curves with no quadratic (quasi-)resonances

Alejandra Gaitan, Victor Lie

Abstract

Building on arXiv:1902.03807, this paper develops a unifying study on the boundedness properties of several representative classes of hybrid operators, i.e. operators that enjoy both zero and non-zero curvature features. Specifically, via the LGC-method, we provide suitable $L^p$ bounds for three classes of operators: (1) Carleson-type operators, (2) Hilbert transform along variable curves, and, taking the center stage, (3) Bilinear Hilbert transform and bilinear maximal operators along curves. All these classes of operators will be studied in the context of hybrid curves with no quadratic resonances. The above study is interposed between two naturally derived topics: i) A prologue providing a first rigorous account on how the presence/absence of a higher order modulation invariance property interacts with and determines the nature of the method employed for treating operators with such a property. ii) An epilogue revealing how several key ingredients within our present study can blend and inspire a short, intuitive new proof of the smoothing inequality that plays the central role in the analysis of the curved version of the triangular Hilbert transform treated in arXiv:2008.10140.

Non-zero to zero curvature transition: Operators along hybrid curves with no quadratic (quasi-)resonances

Abstract

Building on arXiv:1902.03807, this paper develops a unifying study on the boundedness properties of several representative classes of hybrid operators, i.e. operators that enjoy both zero and non-zero curvature features. Specifically, via the LGC-method, we provide suitable bounds for three classes of operators: (1) Carleson-type operators, (2) Hilbert transform along variable curves, and, taking the center stage, (3) Bilinear Hilbert transform and bilinear maximal operators along curves. All these classes of operators will be studied in the context of hybrid curves with no quadratic resonances. The above study is interposed between two naturally derived topics: i) A prologue providing a first rigorous account on how the presence/absence of a higher order modulation invariance property interacts with and determines the nature of the method employed for treating operators with such a property. ii) An epilogue revealing how several key ingredients within our present study can blend and inspire a short, intuitive new proof of the smoothing inequality that plays the central role in the analysis of the curved version of the triangular Hilbert transform treated in arXiv:2008.10140.
Paper Structure (71 sections, 19 theorems, 557 equations, 4 figures)

This paper contains 71 sections, 19 theorems, 557 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A unified theory: context
  3. A brief outlook
  4. Main results
  5. Prologue: A unified perspective on the zero/non-zero curvature paradigm
  6. The two end-points of the spectrum: a philosophical overview
  7. A comparative discussion
  8. Conclusion
  9. The Bilinear Hilbert transform $H_{a,b}^{\alpha}$, $\alpha\in \mathbb{R}_{+}\setminus\{1,2\}$: the initial decomposition of its associated form $\Lambda=\Lambda^{NSI}\,+\,\Lambda^{LO}\,+\,\Lambda^{NS}\,+\,\Lambda^{S}$
  10. The spatial scale decomposition
  11. The phase-dependent decomposition
  12. The phase-space decomposition
  13. The analysis of the stationary term $\Lambda^{S}$: Implementation of the LGC methodology.
  14. Statements of the key results
  15. The $L^2\times L^2\times L^\infty$ bound: Proof of Proposition \ref{['L2mdecay']}
  16. ...and 56 more sections

Key Result

Theorem 1.1

Let $a\in\mathbb{R}\setminus\{-1\}$, $b\in \mathbb{R}$ andFor a discussion on the necessity of imposing the restriction $\alpha\not=2$ please see Observation alfa2 and Section H2discussion below.$\alpha\in(0,\,\infty)\setminus\{1,2\}$. Then, taking $\gamma(t):=a t\,+\,b t^{\alpha}$, we have that the obeys the bound where here $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, $p,\,q\geq 1$ and $r>\frac{2}{3}

Figures (4)

  • Figure 1: Few heavy incidences: if two curves corresponding to two well separated values of $x's$ are incident to the same rectangle $I_l\times J_w$ then they must intersect transversally.
  • Figure 2: Few heavy incidences: due to the non-zero curvature in the $t$ parameter, the presence of the vertical shift does not alter the transversal intersection property for two curves arising from well separated values of $x$.
  • Figure 3: Many heavy incidences are possible: due to the zero curvature in the $t$ parameter, the presence of the vertical shift allows non-transversal intersection for curves arising from well separated values of $x$.
  • Figure 4: The LGC localization is blurring the relational time-frequency Heisenberg localization of a generalized wave-packet thus destroying the almost orthogonality between the thinner tubes.

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • ...and 23 more