Curvature-Related Problems in Harmonic Analysis

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This mathematics research project by Betsy Stovall will study curvature-related problems arising in Euclidean harmonic analysis and their applications to dispersive partial differential equations. This work will encompass two directions. One is to prove new, curvature-independent bounds for generalized Radon transforms defined by averages, linear and multilinear, along families of curves in Euclidean space. Another direction will be to prove new estimates, both linear and bilinear, for the restriction of the Fourier transform to hypersurfaces and apply these and existing estimates to certain dispersive partial differential equations. The study of curvature-independent bounds for translation-invariant generalized Radon transforms and Fourier restriction operators has been a popular and fruitful line of research during recent years. Work performed under this proposal will generalize, extend, and sharpen some of these results. The particular case of averages on curves has been an important model for averages on higher dimensional manifolds, and there is a strong potential that Stovall's work to prove curvature-independent estimates will facilitate future research in these directions. This mathematics research project by Betsy Stovall is in the general area of harmonic analysis, with a focus on the study of so-called generalized Radon transforms and Fourier restriction operators, and their relation with the geometric notion of curvature. Both types of operators have been a significant focus of research in the harmonic analysis community for many years, and their study is part of a broader program to understand the effects of curvature on some operators that arise quite naturally in engineering and physics. For example, one of the specific classes of operators whose study is proposed, a family of restricted X-ray transforms, is related to the (unrestricted) X-ray transform, which is a fundamental tool used in medical imaging. In recent years, research into Fourier restriction operators has been particularly active. This activity is due both to intrinsic interest in these operators and because their study has led to important advances in the understanding of equations arising in fields such as quantum mechanics and optics. The inquiries proposed by Stovall address both of these motivations. As part of this project, Stovall will make a dedicated effort to increase the participation of students and junior scientists from under-represented groups.