Problems of Harmonic Analysis related to finite type hypersurfaces in R^3, and Newton polyhedra

TL;DR

This paper reviews recent results on harmonic analysis related to finite type hypersurfaces in R^3, emphasizing the role of Newton polyhedra and guided by the foundational ideas of Elias M. Stein.

Contribution

It provides an overview of joint work on harmonic analysis problems involving finite type hypersurfaces and Newton polyhedra, including detailed proofs and methodological insights.

Findings

Characterization of harmonic analysis problems on finite type hypersurfaces

Development of techniques involving Newton polyhedra

Connections to foundational ideas of E.M. Stein

Abstract

This article, which grew out of my lecture at the conference "Analysis and Applications: A Conference in Honor of Elias M. Stein" in May 2011, is intended to give an overview on a collection of results which have been obtained jointly with I.I. Ikromov, and in parts also with M. Kempe, and at the same time to give a kind of guided tour through the rather comprehensive proofs of the major results that I shall address. All of our work is highly influenced by the pioneering ideas developed by E.M. Stein.