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A new approach to the Fourier extension problem for the paraboloid

Camil Muscalu, Itamar Oliveira

Abstract

We propose a new approach to the Fourier restriction conjectures. It is based on a discretization of the Fourier extension operators in terms of quadratically modulated wave packets. Using this new point of view, and by combining natural scalar and mixed norm quantities from appropriate level sets, we prove that all the $L^{2}$-based $k$-linear extension conjectures are true up to the endpoint for every $1 \leq k \leq d+1$ if one of the functions involved is a full tensor. We also introduce the concept of \textit{weak transversality}, under which we show that all conjectured $L^{2}$-based multilinear extension estimates are still true up to the endpoint provided that one of the functions involved has a weaker tensor structure, and we prove that this result is sharp. Under additional tensor hypotheses, we show that one can improve the conjectured threshold of these problems in some cases. In general, the largely unknown multilinear extension theory beyond $L^{2}$ inputs remains open even in the bilinear case; with this new point of view, and still under the previous tensor hypothesis, we obtain the near-restriction target for the $k$-linear extension operator if the inputs are in a certain $L^{p}$ space for $p$ sufficiently large. The proof of this result is adapted to show that the $k$-fold product of linear extension operators (no transversality assumed) also ``maps near restriction" if one input is a tensor. Finally, we exploit the connection between the geometric features behind the results of this paper and the theory of Brascamp-Lieb inequalities, which allows us to verify a special case of a conjecture by Bennett, Bez, Flock and Lee.

A new approach to the Fourier extension problem for the paraboloid

Abstract

We propose a new approach to the Fourier restriction conjectures. It is based on a discretization of the Fourier extension operators in terms of quadratically modulated wave packets. Using this new point of view, and by combining natural scalar and mixed norm quantities from appropriate level sets, we prove that all the -based -linear extension conjectures are true up to the endpoint for every if one of the functions involved is a full tensor. We also introduce the concept of \textit{weak transversality}, under which we show that all conjectured -based multilinear extension estimates are still true up to the endpoint provided that one of the functions involved has a weaker tensor structure, and we prove that this result is sharp. Under additional tensor hypotheses, we show that one can improve the conjectured threshold of these problems in some cases. In general, the largely unknown multilinear extension theory beyond inputs remains open even in the bilinear case; with this new point of view, and still under the previous tensor hypothesis, we obtain the near-restriction target for the -linear extension operator if the inputs are in a certain space for sufficiently large. The proof of this result is adapted to show that the -fold product of linear extension operators (no transversality assumed) also ``maps near restriction" if one input is a tensor. Finally, we exploit the connection between the geometric features behind the results of this paper and the theory of Brascamp-Lieb inequalities, which allows us to verify a special case of a conjecture by Bennett, Bez, Flock and Lee.

Paper Structure

This paper contains 26 sections, 26 theorems, 400 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Discrete models
  3. The linear model ($k=1$)
  4. The multilinear model ($2\leq k\leq d+1$)
  5. Transversality versus weak transversality
  6. Our approach and its building blocks
  7. Proof of Proposition \ref{['prop1apr2922']} - Strichartz/Tomas-Stein for $E_{d}$ ($k=1$, $p=2$)
  8. Proof of Proposition \ref{['prop2apr2922']} - Conjecture \ref{['restriction']} for $E_{1}$ ($k=1$, $d=1$, $p=4$)
  9. Proof of Proposition \ref{['prop3apr2922']} - Conjecture \ref{['klinear']} for $ME_{2,1}$ ($k=2$, $d=1$)
  10. Case $k=1$ of Theorem \ref{['mainthmpaper']}
  11. Case $2\leq k\leq d+1$ of Theorem \ref{['mainthmpaper']}
  12. Understanding the core ideas in the $k=d=2$ case
  13. The general argument
  14. The endpoint estimate of the case $k=d+1$ of Theorem \ref{['mainthmpaper']}
  15. Improved $k$-linear bounds for tensors
  16. ...and 11 more sections

Key Result

Theorem 1.5

Conjectures restriction and klinear hold up to the endpoint if one (any) of the functions involved is a full tensorA function $g$ in $d$ variables is a full tensor if it can be written as g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d}).We refer the reader to Satoru and Tanaka for oth

Figures (7)

  • Figure 1: A choice of normal vectors to the caps parametrized by $U_{j}$ via $x\mapsto |x|^{2}$.
  • Figure 2: The phase-space portrait of $\varphi_{n,m}$
  • Figure 3: Transversality versus weak transversality
  • Figure 4: Unveiling the geometric features of the problem when $d=2$. The cubes we find from Theorem \ref{['stability-jan1623']} are weakly transversal, which gives us access to our earlier results.
  • Figure 5: Cases $k=3$ and $k=4$ when $d=3$
  • ...and 2 more figures

Theorems & Definitions (110)

  • Conjecture 1.1
  • Conjecture 1.2: Benn1
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 100 more