Perturbed Brascamp-Lieb inequalities and application to Parsell-Vinogradov systems

Abstract

In this thesis, we study the perturbed Brascamp-Lieb inequalities and its applications in translation-dilation systems. We prove the endpoint perturbed Brascamp-Lieb inequalities using polynomial partition techniques. We also look at the Parsell-Vinogradov system and verify the Brascamp-Lieb condition holds in its decoupling approach. As a corollary of this and the work of Guo, the main conjecture about the system is true in dimension $2$ and can be proved by the decoupling approach.