The quest to find the sharp forms of functional inequalities has always been a beautiful and challenging theme in analysis. Among the most celebrated achievements of the last century one could certainly include the sharp forms of the Hausdorff-Young inequality for the Fourier transform, Young’s inequality for convolutions, the Hardy-Littlewood-Sobolev inequality and the classical Sobolev embeddings.
In this talk we will discuss a few sharp inequalities related to the Fourier transform, within two themes: Fourier restriction theory and Fourier uncertainty principles. In the first theme, we will be mainly concerned with the classical restriction setup for quadratic surfaces (spheres, paraboloids, cones and hyperboloids) arising from the seminal work of Strichartz of 1977. Related questions involving mixed norms and stability issues will also be addressed. In the second theme, we will address the problem of prescribing the sign of a function and its Fourier transform at infinity, and doing this in an optimal way (in an appropriate sense). This phenomenon was introduced by Bourgain, Kahane and Clozel in 2010 under the name of "sign Fourier uncertainty", and brings interesting connections to the sphere packing problem.