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Maximal averages and non-transversality

Jin Bong Lee, Juyoung Lee, Jeongtae Oh, Sewook Oh

TL;DR

This work analyzes how transversality shapes maximal averages over analytic hypersurfaces in $\mathbb{R}^d$, establishing a dichotomy: non-transversal points yield $L^p$ bounds for all $p>2$ independent of surface Fourier decay, while transversal regions tie $L^p$ bounds to Fourier decay with rate $|\widehat{d\mu}(\xi)| \lesssim |\xi|^{-1/p}$ for $p>2$. The authors develop an analytic, $o$-minimal–based surface-decomposition strategy around non-transversal points, and combine this with $L^2$- and smoothing estimates to control the maximal operators; for transversal regions, they relate maximal bounds to sublevel-set geometry and Fourier decay via a stationary-set method. They reformulate the Iosevich–Sawyer–Stein conjecture in the analytic setting, proving equivalence between maximal-bounds and decay assertions and resolving the refined conjecture for $p_{cr}=2$ in several cases. The results clarify the distinct roles of local degeneracy and global oscillation, advancing understanding of maximal averages in harmonic analysis and connecting geometric non-transversality with harmonic-analytic obstructions and sublevel-set techniques. Overall, the paper provides a robust framework to separate geometric non-degeneracy effects from Fourier-analytic decay in the study of maximal operators on analytic surfaces, with potential implications for related restriction-type problems and sublevel-set estimates.

Abstract

We investigate the $L^p$ mapping properties of maximal functions associated with analytic hypersurfaces in $\mathbb R^d$, with a particular emphasis on the role of transversality. Around points that are not transversal, we show that the associated maximal function is bounded on $L^p(\mathbb R^d)$ for all $p>2$, regardless of the decay of the Fourier transform of surface measures. In contrast, away from non-transversal points, we prove that $L^p$ bounds for the maximal operator imply that the Fourier transform of the surface measure decays at rate $1/q$ for $q>p$. Combining these two regimes, we demonstrate that the conjecture of Stein and Iosevich-Sawyer on maximal functions could be re-formulated, in the analytic setting, by restricting attention to transversal points. Moreover, our result completely settles the refined form of the conjecture for certain cases.

Maximal averages and non-transversality

TL;DR

This work analyzes how transversality shapes maximal averages over analytic hypersurfaces in , establishing a dichotomy: non-transversal points yield bounds for all independent of surface Fourier decay, while transversal regions tie bounds to Fourier decay with rate for . The authors develop an analytic, -minimal–based surface-decomposition strategy around non-transversal points, and combine this with - and smoothing estimates to control the maximal operators; for transversal regions, they relate maximal bounds to sublevel-set geometry and Fourier decay via a stationary-set method. They reformulate the Iosevich–Sawyer–Stein conjecture in the analytic setting, proving equivalence between maximal-bounds and decay assertions and resolving the refined conjecture for in several cases. The results clarify the distinct roles of local degeneracy and global oscillation, advancing understanding of maximal averages in harmonic analysis and connecting geometric non-transversality with harmonic-analytic obstructions and sublevel-set techniques. Overall, the paper provides a robust framework to separate geometric non-degeneracy effects from Fourier-analytic decay in the study of maximal operators on analytic surfaces, with potential implications for related restriction-type problems and sublevel-set estimates.

Abstract

We investigate the mapping properties of maximal functions associated with analytic hypersurfaces in , with a particular emphasis on the role of transversality. Around points that are not transversal, we show that the associated maximal function is bounded on for all , regardless of the decay of the Fourier transform of surface measures. In contrast, away from non-transversal points, we prove that bounds for the maximal operator imply that the Fourier transform of the surface measure decays at rate for . Combining these two regimes, we demonstrate that the conjecture of Stein and Iosevich-Sawyer on maximal functions could be re-formulated, in the analytic setting, by restricting attention to transversal points. Moreover, our result completely settles the refined form of the conjecture for certain cases.
Paper Structure (21 sections, 26 theorems, 193 equations)

This paper contains 21 sections, 26 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Maximal estimates associated to non-transversality
  3. Maximal estimates associated to non-transversality
  4. Reformulation of Iosevich--Sawyer--Stein conjecture
  5. Preliminaries
  6. The Ł ojasiewicz inequality and a decomposition of analytic hypersurfaces
  7. $o$-minimal expansions of $\mathbb{R}$
  8. Proof of Proposition \ref{['prop_key']}: $L^2$ bounds
  9. Proof of \ref{['ineq:pc']}
  10. Proof of \ref{['ineq:po']}
  11. Proof of Proposition \ref{['prop_smoothing']}: Smoothing estimates
  12. $2^{\ell_m}<\lambda^{-\frac{1}{2}}$ for some $m=2,\dots, k-1$
  13. $2^{\ell_m} \geq \lambda^{-\frac{1}{2}}$ for all $m=2, \dots, k-1$
  14. Proof of Lemma \ref{['lem_out']}
  15. Proof of \ref{['L2']}: $L^\infty$-bounds for $\mathfrak{m}_{ \vec{\ell}, \lambda}^{\text{out}}$
  16. ...and 6 more sections

Key Result

Theorem 1.3

Let $d\ge2$ and $\Gamma$ be an analytic hypersurface. Suppose that a point $(y_{nt}, \gamma(y_{nt}))$ is non-transversal. If $\psi$ has a sufficiently small support around $y_{nt}$, then $M_{\Gamma}[\psi]$ is bounded on $L^p(\mathbb{R}^d)$ for $p>2$.

Theorems & Definitions (42)

  • Conjecture 1.1: IS2, IKM
  • Definition 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 2.1: Loj1965, Ł ojasiewicz inequality
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • ...and 32 more