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 ),

We use the method of induction-on-scales to prove certain diffeomorphism-invariant nonlinear BrascampLieb inequalities. We provide applications to multilinear convolution inequalities and the restriction theory for the Fourier transform, extending to higher dimensions recent work of Bejenaru-Herr-Tataru and Bennett-Carbery-Wright.

Our main result is that for d = 1, 2 the classical Strichartz norm e is f Lassociated to the free Schrödinger equation is nondecreasing as the initial datum f evolves under a certain quadratic heat flow.

We prove a sharp bilinear estimate for the wave equation from which we obtain the sharp constant in the Strichartz estimate which controls the L 4 t,x (R 5+1 ) norm of the solution in terms of the energy. We also characterise the maximisers.

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