Stefan Buschenhenke

Title: Maximal operators associated to hypersurfaces in $\mathbb R^3$

Abstract: We discuss $L^p$ bounds for maximal operators associated to two-dimensional surfaces in $\R^3$ and present an overview on recent results. In certain cases, depending on the so-called height and on the number of vanishing principal curvatures, this leads to the study of a certain class of Fourier multipliers, which are similar to the cone multiplier, but in addition these multipliers involve an oscillatory part. We discuss properties, bounds and conjectures for this new class of operators. The conjectured $L^4$ bound which is of interest for the maximal operators seems to be of similar difficulty as for the cone multiplier, but we can prove a analogous result for a lower dimensional multiplier.
This is joint work with Spyros Dendrinos, Isroil Ikromov and Detlef Müller.