Problems in Euclidean harmonic analysis related to the geometry of curves and surfaces

• Applicant: Dr. Spyridon Dendrinos, since 1/2014
• Subject Area: Mathematics
• Term: from 2013 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 230032422

• Final Report Year: 2016

Final Report Abstract

The research focused on fundamental questions in harmonic analysis that are central to the theory of the restriction of the Fourier transform and integral transforms, especially convolution with measures supported of lower dimensional manifolds in Euclidean space and the restricted X-ray transform. These operators underpin several modern day applications of an inverse problem nature; medical imaging (computed tomography, positron emission tomography, cone beam computed tomography etc.) being a widely known example. The questions are basic pure-mathematical and aim to quantify the precise amount of smoothing that occurs under these operations. It is hoped that this could provide new ideas or points of view that may in time lead to improvements in the algorithms currently used. The first of the two articles that resulted from this funded research focused on an inequality of a geometric nature that has been the key ingredient in the proofs of Fourier restriction theorems and also of theorems that determine the smoothing (more precisely improvment of integrability in the scale of Lebesgue spaces) effect of both convolution and the restricted X-ray transform. This was the first instance that this geometric inequality has been studied under purely geometric conditions and without relying on algebraic properties of the underlying curves (polynomial, rational, monomial-like etc.). The link of this geometric inequality to convolution with measures supported on curves and the restricted X-ray transform is also explicitely given. In the second article the attention was turned to cases were the underlying manifonld, on which the convolution measure is supported, is a hypersurface in Euclidean space. Here the relation between weighted and unweighted results becomes more intricate and that leads to altogether different strategies of proof having to be considered. The article considers mixed-homogeneous polynomials as a major step in this direction. In doing so it also disproves an almost 20 year old conjecture of Iosevich, Sawyer and Seeger.

Publications

• Uniform bounds for convolution and restricted X-ray transforms along degenerate curves, J. Funct. Anal. 268 (2015), no. 3, 585–633
  Dendrinos, Spyridon & Stovall, Betsy