\(L^p\) boundedness of maximal averages over hypersurfaces in \(\mathbb R^3\) (Q2839383)

Let \(Q\) be a smooth hypersurface in \(\mathbb{R}^3\) and \(q_0 \in Q\). For a small cutoff function \(\phi\) supported near \(q_0\), the maximal operator \(M\) is defined by NEWLINE\[NEWLINEMf(x) = \sup_{t>0} \big| \int_{Q} f(x-tq)\phi(q) \, d\sigma(q) \big|,NEWLINE\]NEWLINE where \(d\sigma\) is the standard surface measure on a smooth hypersurface \(Q\) in \(\mathbb{R}^3\).NEWLINENEWLINEIn the cases that \(Q\) is the Euclidean unit sphere or has nonvanishing Gaussian curvature in \(\mathbb{R}^{n}\), \(n \geq 3\), the \(L^p\) boundedness of the maximal operator \(M\) was proved by \textit{E. M. Stein} [Proc. Natl. Acad. Sci. USA 73, 2174--2175 (1976; Zbl 0332.42018)] and \textit{A. Greenleaf} [Indiana Univ. Math. J. 30, 519--537 (1981; Zbl 0517.42029)], respectively.NEWLINENEWLINEThere are also further related results in the case that \(Q\) has at least one nonvanishing principal curvature and in the case of convex surfaces of finite line type.NEWLINENEWLINEThe author proves sharp \(L^p\) boundedness of maximal averages over hypersurfaces in \(\mathbb{R}^3\) by using facts about adapted coordinate systems in two variables and some methods of \textit{C. D. Sogge} and \textit{E. M. Stein} [Invent. Math. 82, 543--556 (1985; Zbl 0626.42009)]. This provides another approach to the recent work of \textit{I. A. Ikromov, M. Kempe} and \textit{D. Müller} [Acta Math. 204, No. 2, 151--271 (2010; Zbl 1223.42011)].