Flow methods in geometric aspects of harmonic analysis.

Lead Research Organisation: University of Birmingham
Department Name: School of Mathematics


This is a project in Euclidean Harmonic Analysis focussing on the application of flow methods in the context of multilinear functional inequalities. The applicability of such methods in harmonic analysis has seen an unexpected surge over the last 5-10 years, with important consequences in areas such as partial differential equations and number theory. The purpose of this project is to refine these methods in various contexts related to the multilinear restriction theory of the Fourier transform. More specifically, the project aims to obtain near-monotonicity statements for the so-called nonlinear Brascamp-Lieb functional, and explore its consequences for the nonlinear Brascamp-Lieb inequality.


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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509590/1 01/10/2016 30/09/2021
1935855 Studentship EP/N509590/1 25/09/2017 30/03/2021 Rory Duncan
Description Brascamp-Lieb Inequalities are a natural generalisation of various well-known multilinear inequalities found in analysis, encompassing Holder's inequality, Young's Convolution inequality, and the Loomis-Whitney inequality, to name a few. In the traditional regime, each Brascamp-Lieb inequality is defined by a collection of linear maps between euclidean spaces and a corresponding collection of exponents, however in applications it is common to encounter variants where these maps are instead nonlinear. The subject of nonlinear Brascamp-Lieb inequalities has many open problems, amongst which is the question of whether or not they are stable under some class of perturbation of the underlying maps. In analogy with known heat-flow monotonicity properties enjoyed by the linear inequalities, we establish that as the input functions undergo a certain smoothing process, the quantity on the left-hand side of a nonlinear Brascamp-Lieb inequality increases, provided that the maps satisfy certain reasonable uniformity conditions. This is a property that allows us to perturb the underlying maps, thereby establishing a stability result for the set of nonlinear Brascamp-Lieb inequalities that hold.

Moving forward, we wanted to study diffeomorphism-invariant alternatives where no uniformity conditions are satisfied, with the thought that this might be a more geometrically natural setting. We found that by introducing a weight to the left-hand side that mitigates local degeneracies in the nonlinear maps, we prove such a nonlinear Brascamp-Lieb inequality for 'quasialgebraic' maps, this being a broad class of maps for which the notion of 'degree' makes sense (polynomial maps counting amongst them). The best constant in these inequalities depends only on the associated dimensions, exponents, and degrees of the underlying maps. Moreover, we may view them as a generalisation of Young's convolution inequality for algebraic groups.
Exploitation Route A good deal of the motivation for the study of nonlinear Brascamp-Lieb inequalities comes from Fourier restriction theory, which itself is a subject that arose from questions in dispersive PDEs. One may sometimes derive from Brascamp-Lieb inequalities multilinear restriction estimates, which in certain contexts may be viewed as a bound on the mass of a pointwise product of solutions to some constant coefficient linear PDEs. It has been a successful strategy to reduce the usual restriction inequalities to multilinear restriction inequalities (e.g. Bourgain-Guth 2012), and so improvement in the multilinear setting has potential benefits for the broader theory. More generally, one may view Brascamp-Lieb and Kakeya inequalities as non-oscilliatory versions of arguably harder problems such as restriction inequalities or oscilliatory integral estimates, and so they are at least, if not vital ingredients to their proof, natural testing grounds for ideas that one may want to apply in these more complicated oscilliatory settings.
Sectors Digital/Communication/Information Technologies (including Software),Energy

URL https://arxiv.org/abs/2001.03141