Fourier Analysis on Bounded and Exterior Domains

University Of New Mexico, Albuquerque NM

Investigators

Abstract

This proposal deals with many of the central questions in Fourier analysis and how they manifest themselves on domains or more generally, Riemannian manifolds with boundary. In the context of the Fourier transform on Euclidean space or the flat torus, topics such as restriction theorems, Strichartz estimates (space-time integrability estimates for wave and Schroedinger equations), and local smoothing inequalities have been subjects of great interest for quite some time. However, many questions remain on how these theories should play out on a bounded or exterior domain. Here it may be appropriate to think of eigenfunctions of the Laplacian, and replace restriction theorems for the torus with integrability estimates on clusters of eigenfunctions. In this regard, the PI intends to explore estimates on the restriction of these clusters to curves in the domain. In the case of Strichartz estimates, the PI and his collaborators are using a parametrix-based approach to obtain results for general domains. Further improvement is expected in certain contexts by making use of a relatively new family of local smoothing estimates. Fourier analysis continues to be a significant factor in the development of both mathematical and physical theories. In particular, it strengthens our understanding of the partial differential equations that arise in mathematical physics. The research pursued here expects to have several applications to the study of wave phenomena and the equations which model it. These investigations should yield insight on how the presence of a hard boundary surface influences the development of waves, a subject of great scientific interest.