Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sparse Bounds for Rough Fourier Integral Operators

Wellars Banzi, Froduald Minani, Solange Mukeshimana, David Rule

Abstract

We proof pointwise bounds for rough Fourier integral operators by the $L^p$ Hardy-Littlewood maximal function. We assume the Fourier integral operators have amplitudes in $L^\infty S^m_ρ$ and phases $\varphi$ such that $\varphi(x,ξ) - x\cdotξ\in L^\infty Φ^1$, and assume a non-degeneracy condition on the matrix $\partial^2_ξ\varphi(x,ξ)$. The pointwise bound holds when \begin{equation*} m < -\fracρ{2}(n-1) - \fracρ{p} - \frac{n}{p}(1-ρ), \end{equation*} which is known to a be sharp condition on $m$ when $ρ=1$, modulo the end-point. Making use of this pointwise bound and known $L^p$ boundedness results when the phase satisfies an additional non-degeneracy condition, we go on to prove sparse form bounds for these operators.

Sparse Bounds for Rough Fourier Integral Operators

Abstract

We proof pointwise bounds for rough Fourier integral operators by the Hardy-Littlewood maximal function. We assume the Fourier integral operators have amplitudes in and phases such that , and assume a non-degeneracy condition on the matrix . The pointwise bound holds when \begin{equation*} m < -\fracρ{2}(n-1) - \fracρ{p} - \frac{n}{p}(1-ρ), \end{equation*} which is known to a be sharp condition on when , modulo the end-point. Making use of this pointwise bound and known boundedness results when the phase satisfies an additional non-degeneracy condition, we go on to prove sparse form bounds for these operators.
Paper Structure (7 sections, 7 theorems, 91 equations, 1 figure)

This paper contains 7 sections, 7 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pointwise Estimates
  4. Low frequency part
  5. Spatially non-localised high frequency part
  6. Spatially localised high frequency part
  7. Conclusion of the proofs

Key Result

Theorem 1.3

Let $T^{\varphi}_{a}$ be a Fourier integral operator defined as in eq:fio with an amplitude $a\in L^\infty S^m_\rho$ and phase function $\varphi$ such that $\varphi(x,\xi) - x\cdot\xi \in L^\infty\Phi^1$. Suppose further that $|\det_{n-1} \partial^2_\xi \varphi(x,\xi)| \geq c > 0$ and for some $r \in [1,2]$, Then we have that there exists a constant $C$ such that and for each bounded and compact

Figures (1)

  • Figure 1: The figure shows the limiting values $m_\rho(r,s)$ of $m$ in Theorem \ref{['thm:sparseform']} in the $(\frac{1}{r},\frac{1}{s'})$-plane. It is a piecewise linear function and the dashed lines depict the boundary of each linear piece. The left axes show the case $\rho\in(0,\frac{1}{2}]$ and the right axes $\rho\in(\frac{1}{2},1]$. The values of $m_\rho(r,s)$ are written on the corners of each linear piece --- only the expression on the corner (1,0) differs between the cases $\rho\leq1/2$ and $\rho>1/2$, and in going from the former case to the later, one linear piece is split into two.

Theorems & Definitions (13)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • ...and 3 more