$L^p$-Estimates for maximal averages along mixed homogeneous hypersurfaces in $\mathbb{R}^{3}$

Abstract

In this paper, we study $L^p$-estimates for maximal averaging operators $\mathcal M$ along hypersurfaces $S$ in $\mathbb{R}^{3}$ which are the graph of a mixed homogeneous function $Φ$ which is analytic away from the origin. The closure of such a surface will pass through the origin, so that the usual transversality condition that had been imposed in many previous works on maximal averages along hypersurfaces will not hold even when $Φ$ is analytic at the origin. As our main result, under mild assumptions which are satisfied for instance for every mixed homogeneous polynomial $Φ,$ we determine the critical Lebesgue exponent $p_c$ for which $\mathcal M$ is $L^p$-bounded for every $p>p_c,$ but unbounded for $p<p_c,$ in terms of multiplicities of the real roots of the Hessian determinant of $Φ.$ It turns out that the study of the contributions by neighborhoods of a certain type of roots is closely related to recent work by Dendrinos, Ikromov and the first and third author on sharp estimates for a maximal averaging operator along a transversal hypersurface of an ``exceptional&#39;&#39; class, whose $L^p$-boundedness had been an open problem for a long time and which has recently been established by means of their new theory of FIO-cone multipliers.