[Paper Review] Estimates for maximal functions associated to hypersurfaces in $\Bbb R^3$ with height $h<2:$ Part I

Stefan Buschenhenke, Spyridon Dendrinos+2 more|arXiv (Cornell University)|Apr 21, 2017

Advanced Harmonic Analysis Research14 references3 citations

TL;DR

This paper establishes $L^p$-boundedness of maximal operators associated with isotropic dilates of smooth hypersurfaces in $\mathbb{R}^3$ with height $h<2$, focusing on surfaces with vanishing principal curvatures at a point and satisfying a transversality condition. Using novel techniques rooted in Newton diagrams, dyadic decompositions, and oscillatory integral estimates, the authors prove that the critical exponent for boundedness remains $p_c = h$, extending prior results from the $h \geq 2$ case to the more delicate $h < 2$ regime.

ABSTRACT

In this article, we continue the study of the problem of $L^p$-boundedness of the maximal operator $M$ associated to averages along isotropic dilates of a given, smooth hypersurface $S$ of finite type in 3-dimensional Euclidean space. An essentially complete answer to this problem had been given about seven years ago by the last named two authors in joint work with M. Kempe for the case where the height h of the given surface is at least two. In the present article, we turn to the case $h&lt;2.$ More precisely, in this Part I, we study the case where $h&lt;2,$ assuming that $S$ is contained in a sufficiently small neighborhood of a given point $x^0\in S$ at which both principal curvatures of $S$ vanish. Under these assumptions and a natural transversality assumption, we show that, as in the case where $h\ge 2,$ the critical Lebesgue exponent for the boundedness of $M$ remains to be $p_c=h,$ even though the proof of this result turns out to require new methods, some of which are inspired by the more recent work by the last named two authors on Fourier restriction to S. Results on the case where $h&lt;2$ and exactly one principal curvature of $S$ does not vanish at $x^0$ will appear elsewhere.

Motivation & Objective

• To extend the $L^p$-boundedness theory of maximal operators associated with isotropic dilates of smooth hypersurfaces in $\mathbb{R}^3$ to the case where the height $h<2$.
• To analyze the behavior of maximal operators near points where both principal curvatures vanish, under a transversality assumption.
• To establish that the critical Lebesgue exponent for $L^p$-boundedness remains $p_c = h$ in the $h<2$ regime, despite the increased analytical complexity.
• To develop new methods—particularly involving Newton diagrams and dyadic decompositions—tailored to the $h<2$ setting, inspired by recent Fourier restriction theory.

Proposed method

• The authors employ a linear change of coordinates to express the hypersurface locally as the graph of a smooth function $\phi$ vanishing to second order at the origin, enabling the use of adapted coordinate systems.
• They utilize Newton diagrams to classify the singularity type of the phase function $\phi$, particularly focusing on $A_2$ and $D_4^+$ singularities, and derive normal forms under linear transformations.
• A dyadic decomposition is applied with respect to distance to the Airy cone and to the principal root jet, allowing localization of the maximal operator into manageable frequency and spatial regions.
• The maximal operator is decomposed into dyadic frequency pieces $\mathcal{M}_k$, and estimates are derived using interpolation between $L^2$ and $L^{1+\varepsilon}$ bounds with polynomial growth in $k$ and $\lambda$.
• Key estimates are obtained via a variation on the Hardy-Littlewood maximal operator and by analyzing contributions from regions near and away from the principal root jet and the Airy cone.
• For $D_4^+$-type singularities, the problem is reduced to an $A_2$-type singularity via a linear change of variables, allowing application of earlier results on $A_2$-type surfaces with small parameters.

Experimental results

Research questions

• RQ1 What is the critical $L^p$-boundedness exponent for maximal operators associated with hypersurfaces in $\mathbb{R}^3$ when the height $h<2$?
• RQ2 How does the $L^p$-boundedness theory change when both principal curvatures vanish at the base point, compared to the case where at least one is non-zero?
• RQ3 Can the critical exponent $p_c = h$ be preserved in the $h<2$ regime, and if so, what new analytical tools are required to prove it?
• RQ4 How do Newton diagrams and dyadic decompositions in the context of oscillatory integrals contribute to estimating maximal operators near degenerate singularities?
• RQ5 To what extent can the $A_2$-type singularity analysis be extended to $D_4^+$-type singularities through coordinate transformations and parameter-dependent estimates?

Key findings

• The critical Lebesgue exponent for $L^p$-boundedness of the maximal operator $\mathcal{M}$ remains $p_c = h$ even when $h < 2$, extending the result from the $h \geq 2$ case.
• For surfaces with a $D_4^+$-type singularity at the base point, the maximal operator $\mathcal{M}_k$ satisfies the estimate $\|\mathcal{M}_k\|_{L^p \to L^p} \lesssim 2^{k(\frac{1}{p} + \delta)}$ for every $p > 3/2$ and $\delta > 0$, uniformly in $k$.
• The $L^2$ operator norm estimate is bounded by $\|\mathcal{M}_k\|_{L^2 \to L^2} \lesssim T^{1/2} 2^{-k/6} \lambda^{-1/3}$, and the $L^{1+\varepsilon}$ norm by $\|\mathcal{M}_k\|_{L^{1+\varepsilon} \to L^{1+\varepsilon}} \lesssim T \lambda^{2/3 + \delta} 2^{k(1/3 - \delta)}$, leading to interpolation-based $L^p$ bounds.
• The proof relies on a reduction to $A_2$-type singularities via a linear change of coordinates, enabling the application of results from earlier work on $A_2$-type surfaces with small parameters.
• The use of dyadic decomposition with respect to distance to the Airy cone allows uniform control over contributions from different spatial and frequency regions, particularly near and away from the cone.
• The authors establish uniform estimates for the maximal operator $\mathcal{M}$ via summation over dyadic frequency blocks, yielding $\|\mathcal{M}\|_{L^p \to L^p} \lesssim \sum_{k=k_0}^\infty 2^{-2k/3} \|\mathcal{M}_k\|_{L^p \to L^p}$, which converges for $p > 3/2$.