# Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in ^{R}3, and Newton Polyhedra *Detlef Müller*

# Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in ^{R}3, and Newton Polyhedra *Detlef Müller*

This chapter presents three sets of problems and explains how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface *S* (here, a smooth, finite type hypersurface in **R**³ with Riemannian surface measure *dσ*). The first problem is a classical question about estimates for oscillatory integrals, and there exists a huge body of results on it, in particular for convex hypersurfaces. The other two problems had first been formulated by Stein: the study of maximal averages along hypersurfaces has been initiated in Stein's work on the spherical maximal function, and also the idea of Fourier restriction goes back to him.

*Keywords:*
harmonic analysis, finite-type hypersurfaces, Newton polyhedra, oscillatory integrals, convex hypersurfaces, maximal averages, spherical maximal function, Fourier restriction