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$L^p$-estimates for FIO-cone multipliers

Stefan Buschenhenke, Spyridon Dendrinos, Isroil A. Ikromov, Detlef Müller

TL;DR

This work develops a framework for $L^p$ estimates of $FIO$-cone multipliers in ${\mathbb R}^3$, by coupling cone localization with translation-invariant Fourier integral operators. It introduces phase-function classes ${\mathcal{F}}^{\kappa_1,\kappa_2,\gamma}$ adapted to the light cone, allowing controlled singularities, and proves bounds in the range $4/3\le p\le 4$ with an improvement factor $R^{-|1/p-1/2|}$ over traditional $SSS$-type methods. Central to the approach are L-box decompositions, Guth–Wang–Zhang semi-norms, and the small mixed-derivative condition (SMD), which together enable scale-by-scale induction and refined almost-orthogonality arguments. The paper also demonstrates a significant application to maximal averages along smooth analytic surfaces in ${\mathbb R}^3$, confirming a conjecture on the critical Lebesgue exponent for a class of exceptional surfaces, while outlining natural higher-dimensional analogues. Overall, the results sharpen cone-multiplier theory by integrating FIO geometry and box-based decompositions, with implications for geometric maximal operators and related harmonic analysis problems.

Abstract

The classical cone multipliers are Fourier multiplier operators which localize to narrow $1/R$-neighborhoods of the truncated light cone in frequency space. By composing such convolution operators with suitable translation invariant Fourier integral operators (FIOs), we obtain what we call FIO-cone multipliers. We introduce and study classes of such FIO-cone multipliers on $\Bbb R^3$, in which the phase functions of the corresponding FIOs are adapted in a natural way to the geometry of the cone and may even admit singularities at the light cone. By building on methods developed by Guth, Wang and Zhang in their proof of the cone multiplier conjecture in $\Bbb R^3,$ we obtain $L^p$-estimates for FIO-cone multipliers in the range $4/3\le p\le 4$ which are stronger by the factor $R^{-|1/p-1/2|}$ than what a direct application of the method of Seeger, Sogge and Stein for estimating FIOs would give. An important application of our theory is to maximal averages along smooth analytic surfaces in $\Bbb R^3.$ It allows to confirm a conjecture on the the critical Lebesgue exponent for a prototypical surface from a small class of ``exceptional'' surfaces, for which this conjecture had remained open.

$L^p$-estimates for FIO-cone multipliers

TL;DR

This work develops a framework for estimates of -cone multipliers in , by coupling cone localization with translation-invariant Fourier integral operators. It introduces phase-function classes adapted to the light cone, allowing controlled singularities, and proves bounds in the range with an improvement factor over traditional -type methods. Central to the approach are L-box decompositions, Guth–Wang–Zhang semi-norms, and the small mixed-derivative condition (SMD), which together enable scale-by-scale induction and refined almost-orthogonality arguments. The paper also demonstrates a significant application to maximal averages along smooth analytic surfaces in , confirming a conjecture on the critical Lebesgue exponent for a class of exceptional surfaces, while outlining natural higher-dimensional analogues. Overall, the results sharpen cone-multiplier theory by integrating FIO geometry and box-based decompositions, with implications for geometric maximal operators and related harmonic analysis problems.

Abstract

The classical cone multipliers are Fourier multiplier operators which localize to narrow -neighborhoods of the truncated light cone in frequency space. By composing such convolution operators with suitable translation invariant Fourier integral operators (FIOs), we obtain what we call FIO-cone multipliers. We introduce and study classes of such FIO-cone multipliers on , in which the phase functions of the corresponding FIOs are adapted in a natural way to the geometry of the cone and may even admit singularities at the light cone. By building on methods developed by Guth, Wang and Zhang in their proof of the cone multiplier conjecture in we obtain -estimates for FIO-cone multipliers in the range which are stronger by the factor than what a direct application of the method of Seeger, Sogge and Stein for estimating FIOs would give. An important application of our theory is to maximal averages along smooth analytic surfaces in It allows to confirm a conjecture on the the critical Lebesgue exponent for a prototypical surface from a small class of ``exceptional'' surfaces, for which this conjecture had remained open.

Paper Structure

This paper contains 29 sections, 26 theorems, 458 equations.

Table of Contents

  1. Introduction
  2. On the proof strategy
  3. Classes of phase functions adapted to the light cone
  4. Stability results and examples
  5. Stability results for the defining estimates of our classes of phases
  6. Rotated and dilated light cones
  7. Examples of phase functions
  8. L-boxes and the main theorem
  9. L-boxes associated to phases in ${\mathcal{F}}^{{\kappa}_1,{\kappa}_2,\gamma,{\mathcal{C}}}$
  10. Decomposition into L-boxes
  11. The small mixed derivative condition (SMD)
  12. Admissible amplitudes
  13. The main result
  14. Preparatory proof steps
  15. Guth-Wang-Zhang semi-norms
  16. ...and 14 more sections

Key Result

Theorem 1.1

For any $\epsilon>0$, there exists a constant $C_\epsilon>0$ such that for any $R\gg 1$ and $\lambda\geq R^\gamma,$

Theorems & Definitions (65)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 55 more