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Maximal estimates for averages over degenerate hypersurfaces

Sewook Oh

TL;DR

The article advances maximal averaging over dilations of a smooth hypersurface S by proving optimal $L^p$ bounds in the Fourier-decay regime $q=1/2$ under a finite-type condition on the defining function $\gamma$. The authors develop a two-pronged strategy: a frequency-localized decomposition that isolates degeneracy and a geometric decomposition away from degeneracy, which is then tackled via cone decoupling techniques. They establish an $L^p$ bound for $p>2$ in the $q=1/2$ case and, for non-flat S, show boundedness on some finite $L^p$, thereby extending previous results by Sogge and Stein. Central to the approach are local smoothing estimates and a careful multiscale decomposition combined with Bourgain–Demeter decoupling for cones, enabling control of oscillatory integrals along degenerate hypersurfaces.

Abstract

We study $L^p$ boundedness of the maximal average over dilations of a smooth hypersurface $S$. When the decay rate of the Fourier transform of a measure on $S$ is $1/2$, we establish the optimal maximal bound, which settles the conjecture raised by Stein. Additionally, when $S$ is not flat, we verify that the maximal average is bounded on $L^p$ for some finite $p$, which generalizes the result by Sogge and Stein.

Maximal estimates for averages over degenerate hypersurfaces

TL;DR

The article advances maximal averaging over dilations of a smooth hypersurface S by proving optimal bounds in the Fourier-decay regime under a finite-type condition on the defining function . The authors develop a two-pronged strategy: a frequency-localized decomposition that isolates degeneracy and a geometric decomposition away from degeneracy, which is then tackled via cone decoupling techniques. They establish an bound for in the case and, for non-flat S, show boundedness on some finite , thereby extending previous results by Sogge and Stein. Central to the approach are local smoothing estimates and a careful multiscale decomposition combined with Bourgain–Demeter decoupling for cones, enabling control of oscillatory integrals along degenerate hypersurfaces.

Abstract

We study boundedness of the maximal average over dilations of a smooth hypersurface . When the decay rate of the Fourier transform of a measure on is , we establish the optimal maximal bound, which settles the conjecture raised by Stein. Additionally, when is not flat, we verify that the maximal average is bounded on for some finite , which generalizes the result by Sogge and Stein.
Paper Structure (10 sections, 18 theorems, 153 equations)

This paper contains 10 sections, 18 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Frequency localization
  3. Proof of \ref{['aa01']}
  4. Decomposition away from degeneracy
  5. Proof of Proposition \ref{['prop:aa0']}
  6. Properties of $\mathfrak a_{\vec{\ell}}$
  7. Proof of Proposition \ref{['prop:main']}
  8. Proof of Theorem \ref{['thm:avl']}
  9. Fourier support of $\mathcal{A}[\mathfrak a_{{\vec{\ell}},\tilde{\mu}}^{\delta_{n+1}}]$
  10. Proof of Proposition \ref{['prop:avl']}

Key Result

Theorem 1.1

Let $\gamma:{\mathbb R}^{d-1}\rightarrow{\mathbb R}$ be a smooth function of finite type. Suppose that Fourierdecay holds for $q=1/2$. Then $\mathcal{M}$ is bounded on $L^p$ for $p>2$.

Theorems & Definitions (28)

  • Conjecture 1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 18 more