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Weighted wave envelope estimates for the parabola

Jongchon Kim, Hyerim Ko

TL;DR

The paper develops weighted square function estimates for the parabola, extending Córdoba–Fefferman-type results to a weighted setting via a multiscale tube/tiling framework around the parabola and a weighted envelope analysis.A central contribution is the introduction of weight-quantifying tubes through $\kappa_{p,H}(U)$ and the demonstration that weighted envelope bounds yield sharp weighted $L^p$ controls for $2\le p\le4$ across several objects, including radial Fourier multipliers and the Schrödinger propagator.The work further establishes fractal local smoothing estimates for the Schrödinger equation with respect to both parabolic and Euclidean fractal measures, identifying precise sufficiency and necessity thresholds for the regularity parameter $\gamma$ as a function of the fractal dimension.Together, these results connect multiscale geometric decompositions, bilinear restriction estimates, and fractal measure theory to produce robust weighted estimates with applications to harmonic analysis and dispersive PDE.

Abstract

In this paper, we extend the Córdoba-Fefferman square function estimate for the parabola to a weighted setting. Our weighted square function estimate is derived from a weighted wave envelope estimate for the parabola. The bounds are formulated in terms of families of multiscale tubes together with weight parameters that quantify the distribution of the weight. As an application, we obtain some weighted L^p-estimates for a class of Fourier multiplier operators and for solutions to free Schrödinger equation.

Weighted wave envelope estimates for the parabola

TL;DR

The paper develops weighted square function estimates for the parabola, extending Córdoba–Fefferman-type results to a weighted setting via a multiscale tube/tiling framework around the parabola and a weighted envelope analysis.A central contribution is the introduction of weight-quantifying tubes through $\kappa_{p,H}(U)$ and the demonstration that weighted envelope bounds yield sharp weighted $L^p$ controls for $2\le p\le4$ across several objects, including radial Fourier multipliers and the Schrödinger propagator.The work further establishes fractal local smoothing estimates for the Schrödinger equation with respect to both parabolic and Euclidean fractal measures, identifying precise sufficiency and necessity thresholds for the regularity parameter $\gamma$ as a function of the fractal dimension.Together, these results connect multiscale geometric decompositions, bilinear restriction estimates, and fractal measure theory to produce robust weighted estimates with applications to harmonic analysis and dispersive PDE.

Abstract

In this paper, we extend the Córdoba-Fefferman square function estimate for the parabola to a weighted setting. Our weighted square function estimate is derived from a weighted wave envelope estimate for the parabola. The bounds are formulated in terms of families of multiscale tubes together with weight parameters that quantify the distribution of the weight. As an application, we obtain some weighted L^p-estimates for a class of Fourier multiplier operators and for solutions to free Schrödinger equation.

Paper Structure

This paper contains 26 sections, 15 theorems, 150 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of \ref{['cor:sqweighted']} and some applications
  3. Weighted wave envelope estimates imply weighted square function estimates
  4. Weighted estimates for a radial Fourier multiplier
  5. Weighted estimates for the Schrödinger equation
  6. Fractal local smoothing estimates relative to parabolic balls
  7. Fractal local smoothing estimates
  8. More examples
  9. A lower bound for the weighted square function estimate
  10. A sharp example beyond interpolation
  11. Weighted wave envelope estimates
  12. Reductions
  13. Decomposition
  14. Proof of \ref{['prop:bil-lp']}
  15. Bilinear restriction estimates
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $2\leq p\leq 4$ and $H:\mathbb{R}^2 \to [0,1]$ be a function. For any function $f$ whose Fourier transform is supported on $N_{R^{-1}} (\mathcal{P})$, we have,

Figures (3)

  • Figure 1: $U\in \mathbb{U}_\tau$, $T\in \mathbb{T}_\tau$ such that $T\subset U$ for $|\tau|=s$.
  • Figure 2: Sufficient (solid) and necessary (dotted) thresholds for \ref{['FLSS_par']}.
  • Figure 3: Blue and black lines denote the upper bounds on from \ref{['eqn:zeta']} and \ref{['eqn:kappaAlpha1']}, respectively, while the dotted line indicates the lower bound from \ref{['eqn:zetaLower']}.

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Example 1: $\alpha$-dimensional weights
  • Example 2: Unit ball
  • Remark 1.3
  • Theorem 1.4
  • Example 3
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • ...and 19 more