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A sharp weighted Fourier extension estimate for the cone in $\mathbb{R}^3$ based on circle tangencies

Alexander Ortiz

TL;DR

The paper tackles sharp weighted Fourier extension estimates for the truncated cone in $\mathbb{R}^3$ with $1$-dimensional weights, advancing partial progress toward the Mizohata--Takeuchi conjecture in this setting. It develops a circle-tangency framework via point-circle duality and a dual lightplank rectangle dictionary, then leverages the Pramanik--Yang--Zahl incidence bound to control tangency configurations and derive weighted $L^2$-type bounds for $E_{\text{Cone}^2}f$. A central achievement is a sharp weighted $L^1$ to $L^2$ transition that yields improved Mizohata--Takeuchi-type estimates for the cone with $R^{1/4+\varepsilon}$-type losses in the 1D weight case, together with a broader discussion of connections to decoupling and conical Fourier averages. The results illuminate the geometry–incidence–PDE pipeline for zero-curvature models and open avenues to refine decoupling-based approaches for cones and spheres in related settings.

Abstract

We apply recent circle tangency estimates due to Pramanik--Yang--Zahl to prove sharp weighted Fourier extension estimates for the cone in $\mathbb{R}^3$ and $1$-dimensional weights. The idea of using circle tangency estimates to study Fourier extension of the cone is originally due to Tom Wolff, who used it in part to prove the first decoupling estimates. We make an improvement to the best known Mizohata--Takeuchi-type estimates for the cone in $\mathbb{R}^3$ and the $1$-dimensional weights as a corollary of our main theorem, where the previously best known bound follows as a corollary of refined decoupling estimates.

A sharp weighted Fourier extension estimate for the cone in $\mathbb{R}^3$ based on circle tangencies

TL;DR

The paper tackles sharp weighted Fourier extension estimates for the truncated cone in with -dimensional weights, advancing partial progress toward the Mizohata--Takeuchi conjecture in this setting. It develops a circle-tangency framework via point-circle duality and a dual lightplank rectangle dictionary, then leverages the Pramanik--Yang--Zahl incidence bound to control tangency configurations and derive weighted -type bounds for . A central achievement is a sharp weighted to transition that yields improved Mizohata--Takeuchi-type estimates for the cone with -type losses in the 1D weight case, together with a broader discussion of connections to decoupling and conical Fourier averages. The results illuminate the geometry–incidence–PDE pipeline for zero-curvature models and open avenues to refine decoupling-based approaches for cones and spheres in related settings.

Abstract

We apply recent circle tangency estimates due to Pramanik--Yang--Zahl to prove sharp weighted Fourier extension estimates for the cone in and -dimensional weights. The idea of using circle tangency estimates to study Fourier extension of the cone is originally due to Tom Wolff, who used it in part to prove the first decoupling estimates. We make an improvement to the best known Mizohata--Takeuchi-type estimates for the cone in and the -dimensional weights as a corollary of our main theorem, where the previously best known bound follows as a corollary of refined decoupling estimates.
Paper Structure (14 sections, 42 theorems, 210 equations, 9 figures)

This paper contains 14 sections, 42 theorems, 210 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Weighted estimates and the Mizohata--Takeuchi conjecture
  3. Decay of Fourier means
  4. Circle tangencies and an overview of the proof of Theorem \ref{['thm:main-fourier-decay']}
  5. Paper organization
  6. Point-circle and rectangle-lightplank duality
  7. Duality constructions
  8. Tangency counting by dual circular maximal estimates
  9. Nearly lightlike separated pairs
  10. Proof of main theorems
  11. From Theorem \ref{['thm:ortiz-sparse-l1']} to Theorem \ref{['thm:main-l2-sparse']}
  12. Refined decoupling and the Mizohata--Takeuchi conjecture
  13. Lemmas of circle geometry
  14. Fourier transform of surface measure on the cone

Key Result

Theorem 1.1

For each $\epsilon > 0$, there is a constant $C_\epsilon$ so the following holds for each $R > 1$. Suppose $X\subset B_R$ is a disjoint union of unit balls in $\mathbb R^3$ that satisfies the 1-dimensional Frostman non-concentration condition Let $\mathbf P(X)$ be the quantity Then the estimate holds for all $f\in L^2(\mathbb Cone^2)$.

Figures (9)

  • Figure 1: Plot of $\gamma_3(\alpha)$
  • Figure 2: Point-circle duality
  • Figure 3: A $\delta,\tau$-rectangle $\Omega$, pictured in gray
  • Figure 4: A snapshot of "the" dual $\delta\times\delta\tau^{-1}\times\delta\tau^{-2}$-lightplank to a $\delta,\tau$-rectangle $\Omega^{(v)}$
  • Figure 5: The four "corner points" of $P^{(o)}_0\cap P^{(o)}_{\frac{\theta}{\sqrt\delta}}$
  • ...and 4 more figures

Theorems & Definitions (90)

  • Conjecture 1.1: Local Mizohata--Takeuchi
  • Conjecture 1.2: Global Mizohata--Takeuchi
  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Definition 1.2
  • Definition 1.3: Optimal spherical Fourier average decay rate
  • Definition 1.4: Optimal conical Fourier average decay rate
  • Theorem 1.3: Erdoğan, 2004
  • ...and 80 more