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

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.