A global approach to nonlinear Brascamp-Lieb inequalities and related topics

Duncan, Jennifer (2022). A global approach to nonlinear Brascamp-Lieb inequalities and related topics. University of Birmingham. Ph.D.

Preview

Duncan2022PhD.pdf
Text - Accepted Version
Available under License All rights reserved.
Download (1MB) | Preview

Abstract

In this thesis, we investigate global nonlinear Brascamp–Lieb inequalities and some related problems in multilinear harmonic analysis. The body of this thesis is split into three parts, the first is concerning the near-monotonicity properties of nonlinear Brascamp–Lieb functionals under heat-flow. We establish a global nonlinear analogy to the heat-flow monotonicity property enjoyed by linear Brascamp–Lieb inequalities, which we use to prove a slight improvement of the local nonlinear Brascamp–Lieb inequality due to Bennett, Bez, Buschenhenke, Cowling, and Flock, as well as a global stability property of the finiteness of nonlinear Brascamp–Lieb inequalities. In the second part we prove a diffeomorphism-invariant weighted nonlinear Brascamp–Lieb inequality for maps thatadmit a certain structure that generalises the class of polynomial maps. Like polynomials, they have a well-defined notion of degree, and the best constant in this inequality depends explicitly on only the degree of these maps, as well as the underlying dimensions and exponents. Lastly, we refine an induction-on-scales method due to Bennett, Carbery, and Tao to prove a global multilinear $L^2$ estimate on oscillatory integral operators in general dimensions.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):