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A degree one Carleson operator along the paraboloid

Lars Becker

TL;DR

The paper develops a time-frequency framework for maximally modulated singular integrals along paraboloids with degree-one modulations, proving $L^p$ bounds in a nontrivial range $ rac{d^2+4d+2}{(d+1)^2}<p<2(d+1)$. Central to the approach is a discretization into tiles, organized into antichains, trees, and forests, with separate analyses yielding weak-type $L^2$ bounds and sparse bounds. Antichains are controlled by a square-function argument; trees yield sparse bounds via Sobolev smoothing and singular-Radon-transform techniques; forests are handled through almost-orthogonality and oscillatory-integral estimates on paraboloids. Interpolation then delivers the $L^p$ bounds, extending Carleson-type convergence results to degree-one operators along paraboloids and highlighting the role of modulation symmetries in guiding the time-frequency analysis. The work advances understanding of degree-one modulation symmetries and provides a robust framework that could inform future extensions to broader subspaces and higher-dimensional manifolds.

Abstract

We prove $L^p$ bounds, $\frac{d^2 + 4d + 2}{(d+1)^2} < p < 2(d+1)$, for maximal linear modulations of singular integrals along paraboloids with frequencies in certain subspaces of $\mathbb{R}^{d+1}$, for $d \geq 2$. This generalizes Carleson's theorem on convergence of Fourier series, and complements a corresponding result by Pierce and Yung with polynomial modulations without linear terms.

A degree one Carleson operator along the paraboloid

TL;DR

The paper develops a time-frequency framework for maximally modulated singular integrals along paraboloids with degree-one modulations, proving bounds in a nontrivial range . Central to the approach is a discretization into tiles, organized into antichains, trees, and forests, with separate analyses yielding weak-type bounds and sparse bounds. Antichains are controlled by a square-function argument; trees yield sparse bounds via Sobolev smoothing and singular-Radon-transform techniques; forests are handled through almost-orthogonality and oscillatory-integral estimates on paraboloids. Interpolation then delivers the bounds, extending Carleson-type convergence results to degree-one operators along paraboloids and highlighting the role of modulation symmetries in guiding the time-frequency analysis. The work advances understanding of degree-one modulation symmetries and provides a robust framework that could inform future extensions to broader subspaces and higher-dimensional manifolds.

Abstract

We prove bounds, , for maximal linear modulations of singular integrals along paraboloids with frequencies in certain subspaces of , for . This generalizes Carleson's theorem on convergence of Fourier series, and complements a corresponding result by Pierce and Yung with polynomial modulations without linear terms.
Paper Structure (27 sections, 22 theorems, 208 equations)

This paper contains 27 sections, 22 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Modulation symmetries
  4. Previous results
  5. Overview of the proof of Theorem \ref{['main Lp']}
  6. Antichains: A square function argument
  7. Trees: Sparse bounds
  8. Forests: Square functions and oscillatory integrals
  9. Limitations and further questions
  10. Reduction to a discretized operator
  11. Outline of the proof of weak type L2-bounds for (8)
  12. Tile organization: Proof of Proposition \ref{['prop organization']}
  13. Antichains: Proof of Proposition \ref{['prop antichain']}
  14. Decomposition of the kernel
  15. The smooth part
  16. ...and 12 more sections

Key Result

Theorem 1

Let $d \geq 2$ and let $m > \frac{d}{2}$. Suppose that $V = \{0\}^d \times \mathbb{R}$ or $V$ is a proper subspace of $\mathbb{R}^{d}\times\{0\}$. Then for all $p$ with there exists $C > 0$ such that for all $m$-Calderón-Zygmund kernels $K$ and all Schwartz functions $f$, we have with $T_V$ as defined in eq operator

Theorems & Definitions (40)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 30 more