Multilinear geometric inequalities

Abstract

In this thesis we explore a multilinear duality principle of Carbery–Hänninen–Valdimarsson for multilinear geometric inequalities arising in Harmonic Analysis. In Chapter 2, we give a brief survey of how this theory manifests for a number of classical inequalities. The material in Chapter 2 is mostly expository except for two original results: Theorem 2.17, which gives an explicit factorisation of the classical Young convolution inequality on R, and Proposition 2.10, establishing the equivalence of two natural formulations of an affine-invariant Finner inequality. The material in Section 2.3 is also new. In Chapter 3, we discuss a certain structural duality between instances of the geometric Brascamp–Lieb inequality, and briefly explore how this manifests on the level of factorisations. Most of the material presented in this chapter is not original research. However, the connection between Lemma 3.2 and Proposition 3.7 in the setting of the geometric Brascamp–Lieb inequality is new. In particular, we give a new proof for a special case of Proposition 3.7, which only relies on the sharpness of the geometric Brascamp–Lieb inequality along with a generalised version of Lemma 3.2. In Chapter 4 we study factorisations due to Guth for the endpoint multilinear Kakeya inequality, and factorisations due to Zhang for a corresponding generalisation to kj -planes. In particular, we uncover interesting connections to classical notions of convex geometry. The main contributions of this chapter stem from the statements and proofs of Theorems 4.7 and 4.13. In particular, these two geometric observations allow us to translate Guth’s visibility theorem, stated below as Theorem 4.33, into statements about convex bodies, which lead us to the formulation of Theorems 4.8 and 4.14. In Chapter 5 we study a sequence of generalised versions of Guth’s directed volume - see Definition 4.3. The main contribution of this chapter is Theorem 5.1, giving a sharp chain of inequalities for such geometric quantities. Furthermore, we conjecture restricted inequalities of the same type, which are optimal for boundaries of parallelotopes and related to a vector-valued generalisation of the classical Maclaurin inequality. In Chapter 6, we prove various special cases of this vector-valued variant of the Maclaurin inequality, and highlight connections to open problems in convex geometry.