Oscillatory integrals and maximal averages over homogeneous surfaces (Q1919618)

This paper is concerned with maximal operators associated to compactly supported smooth measures on hypersurfaces of finite type. To be more precise, a hypersurface \(S\) in \({\mathbb R}^n\) is said to be of finite type if the order of contact of the tangent lines to \(S\) is bounded. Given a smooth surface measure \(\sigma\) on \(S\), we consider the dilates \(\delta_t\sigma\) given by \((\delta_t\sigma)(u)=\sigma(u\circ D_t)\) for any bounded continuous function \(u\) on \({\mathbb R}^n\), where \(D_tx=tx\), and the associated maximal operator \({\mathcal M}\), given by \({\mathcal M}f=\sup_{t>0} | (\delta_t\sigma)*f| \) for compactly supported continuous functions \(f\) on \({\mathbb R}^n\). Estimates of the form \(\| {\mathcal M}f\|_p\leq C\| f\|_p\) for the Lebesgue \(L^p\) norms are studied, for \(p\) in the range \((1,\infty)\). The first main result is that, if such an inequality holds, then \(x\mapsto d(x,H)^{-1/p}\in L^1_{\text{loc}}(S)\) for all hyperplanes \(H\) not passing through the origin (here \(d\) denotes the Euclidean distance); this does not require the finite type assumption. Conversely, if \(S\) is of the form \(x_n=\Phi(x_1,\ldots,x_{n-1})+C\), where \(\Phi\) is homogeneous and \(\Phi^{-1/p_0}\in L^1_{\text{loc}}({\mathbb R}^{n-1})\), then \(\| {\mathcal M}f\|_p\leq C\| f\|_p\) holds if \(p>p_0\geq 2\). Here the finite type assumption is definitely needed.