We prove d-linear analogues of the classical restriction and Kakeya conjectures in R d . Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variablecoefficient problems and the so-called "joints" problem, as well as presenting some n-linear analogues for n < d.

Abstract. We consider the Brascamp-Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse the inequalities from a structural perspective. Our main tool is a monotonicity formula for positive solutions to heat equations in linear and multilinear settings, which was first used in this type of setting by Carlen, Lieb, and Loss [CLL]. In that paper, the heat flow method was used to obtain the rank one case of Lieb's fundamental theorem concerning exhaustion by gaussians; we extend the technique to the higher rank case, giving two new proofs of the general rank case of Lieb's theorem.

We establish a diffeomorphism-invariant generalisation of the classical Loomis-Whitney inequality in R n . As a consequence we obtain a sharp trilinear restriction theorem for the Fourier transform in three dimensions.

Abstract. A criterion is established for the validity of multilinear inequalities of a class considered by Brascamp and Lieb, generalizing well-known inequalties of Rogers and Hölder, Young, and Loomis-Whitney.

We prove that the best constant in the general Brascamp-Lieb inequality is a locally bounded function of the underlying linear transformations. As applications we deduce certain very general Fourier restriction, Kakeya-type, and nonlinear variants of the Brascamp-Lieb inequality which have arisen recently in harmonic analysis. m j=1 ker L j = {0}.In [30] Lieb proved that BL(L, p) is exhausted by centred gaussian inputs f j (x) = exp(−π A j x, x ),