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In this article, we continue the study of $L^p$-boundedness of the maximaloperator $\mathcal M_S$ associated to averages along isotropic dilates of agiven, smooth hypersurface $S$ in 3-dimensional Euclidean space. We focus hereon small surface-patches near a given point $x^0$ exhibiting singularities oftype $\mathcal A$ in the sense of Arnol'd at this point; this is the situationwhich had yet been left open. Denoting by $p_c$ the minimal Lebesgue exponentsuch 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 asmall subclass), by means of quantities which can be determined from associatedNewton polyhedra. Besides the well-known notion of height at $x^0,$ a newquantity, which we call the effective multiplicity, turns out to play a crucialrole here. We also state a conjecture on how the critical exponent $p_c$ mightbe determined by means of a geometric measure theoretic condition, whichmeasures in some way the order of contact of arbitrary ellipsoids with $S,$even for hypersurfaces in arbitrary dimension, and show that this conjectureholds indeed true for all classes of 2-hypersurfaces $S$ for which we havegained an essentially complete understanding of $\mathcal M_S$ so far. Ourresults lead in particular to a proof of a conjecture by Iosevich-Sawyer-Seegerfor arbitrary analytic 2-surfaces.

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