Estimates for maximal functions associated to hypersurfaces in $\Bbb R^3$ with height $h<2:$ Part II -- A geometric conjecture and its proof for generic 2-surfaces

TL;DR

This paper advances understanding of the $L^p$-boundedness of maximal operators associated with hypersurfaces in 3D, especially near singular points, by identifying critical exponents through geometric and Newton polyhedron analysis.

Contribution

It determines the critical Lebesgue exponent $p_c$ for $L^p$-boundedness of maximal operators on analytic 2-surfaces with singularities, introducing the effective multiplicity and confirming a geometric conjecture.

Findings

Identified $p_c$ for all analytic surfaces of type $\\mathcal{A}$.

Introduced the effective multiplicity as a key quantity.

Proved the conjecture for all classes of 2-hypersurfaces with known $\\mathcal{M}_S$ understanding.

Abstract

In this article, we continue the study of $L^p$-boundedness of the maximal operator $\mathcal M_S$ associated to averages along isotropic dilates of a given, smooth hypersurface $S$ in 3-dimensional Euclidean space. We focus here on small surface-patches near a given point $x^0$ exhibiting singularities of type $\mathcal A$ in the sense of Arnol'd at this point; this is the situation which had yet been left open. Denoting by $p_c$ the minimal Lebesgue exponent such that $\mathcal M_S$ is $L^p$-bounded for $p>p_c,$ we are able to identify $p_c$ for all analytic surfaces of type $\mathcal A$ (with the exception of a small subclass), by means of quantities which can be determined from associated Newton polyhedra. Besides the well-known notion of height at $x^0,$ a new quantity, which we call the effective multiplicity, turns out to play a crucial role here. We also state a conjecture on how the critical exponent $p_c$ might be determined by means of a geometric measure theoretic condition, which measures in some way the order of contact of arbitrary ellipsoids with $S,$ even for hypersurfaces in arbitrary dimension, and show that this conjecture holds indeed true for all classes of 2-hypersurfaces $S$ for which we have gained an essentially complete understanding of $\mathcal M_S$ so far. Our results lead in particular to a proof of a conjecture by Iosevich-Sawyer-Seeger for arbitrary analytic 2-surfaces.