Damped oscillatory integrals and maximal operators (Q2508748)

The analogue of Stein's classical theorems on spherical means has been much studied for maximal operators of the type \[ Mg(x)= \sup_{t> 0}\,\Biggl|\int_S g(x- ty)\psi(y)\,d\sigma(y)\Biggr| \] taken over an analytical hypersurface \(S\) in \(\mathbb{R}^{n+1}\). The smallest exponent \(p= p(S)\) such that the inequality \[ \| Mg\|_{L^p(S)}\leq C\| g\|_{L^p(S)} \] holds for all \(g\in C^\infty_0(\mathbb{R}^{n+1})\) is of crucial interest. In example, if \(S\) is strictly convex, it is known that \(p(S)\) is finite. The present work employs the curvature tensor. The inequality is proved under a rather general technical assumptions formulated in terms of a so-called mitigating factor, expressed through the first and second fundamental forms of the surface. An estimate of rapidly oscillation integrals is essential, the ``mitigating factors'' appearing as damping weights.