2022/09/15 by Stefan Buschenhenke, Buschenhenke, Stefan, Isroil A. Ikromov +3
Mathematics · #42B25 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2209.07352
openalex publication_date 2022/09/15 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
In this article, we continue the study of Lp-boundedness of the maximal operator \mathcal MS 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 x0 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 pc the minimal Lebesgue exponent such that \mathcal MS is Lp-bounded for p>pc, we are able to identify pc 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 x0, 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 pc 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 MS so far. Our results lead in particular to a proof of a conjecture by Iosevich-Sawyer-Seeger for arbitrary analytic 2-surfaces.