Two geometric inequalities in harmonic analysis

We study two geometric inequalities in harmonic analysis.In the first part we study the Brascamp-Lieb inequality. We re- examine several of the approaches that have yielded results for this inequality and use them to derive new results. Specifically we prove an inequality involving the Hessian of the optimal transport map and use it to derive the generalised Brascamp-Lieb and reverse Brascamp-Lieb inequality with the methods of Barthe. Also, we extend the heat flow methods from Carlen, Lieb and Loss to give the form of all optimisers for the Brascamp-Lieb inequality and we use the induction on dimension method of Bennett, Carbery, Christ and Tao to prove a Brascamp-Lieb inequality for finite fields. Finally, we study the set of LP- indices where the Brascamp-Lieb inequality holds and give alternative ways of describing it in several situations.In the second part we study a multilinear analogue of fractional integration which has been studied in one form by Drury. We give the L bounds for it and find the optimal constant for this bound in the case with the most symmetries. We also determine all functions which are optimisers for this inequality. Finally, we study an analogue of this form which corresponds to the Hilbert transform. Here the finiteness of the form depends on cancellation properties in the kernel and we show how to define the form in terms of distributions. Then we prove L bounds for that form