The goal of this talk is to present a new point of view to study the Fourier restriction/extension problems. In a nutshell, it allows us to prove that the linear and multilinear L^2-based Fourier extension conjectures for the paraboloid are true (including the endpoint in the d-linear case) if one of the input functions is a full tensor. In the multilinear case, this result holds under the assumption that the domains of the inputs are weakly transversal, rather than transversal. The method extends to obtain “near-restriction” multilinear estimates (i.e., beyond the L^2-based theory) with and without transversality. We will also go over some applications of the tools used, which includes establishing an endpoint Restriction-Brascamp-Lieb conjecture for certain submanifolds (a problem posed by Bennett, Bez, Flock and Lee). This is joint work with Camil Muscalu.

Fourier optimization problems ask us to optimize a quantity of interest, given restrictions on a function and its Fourier transform. Meanwhile, uncertainty principles give us a limit of how far we can go. These frameworks often have applications to other fields. For instance, studying integers and prime numbers represented by quadratic forms is a classical problem in Number Theory, going back to Fermat. We’ll discuss a Fourier analysis approach to this problem that allows us to obtain new estimates in the theory, based on joint work with Andrés Chirre.

In this talk we will discuss a family of extremal problems that arise from the problem of determining the operator norm of certain embeddings between weighted Paley-Wiener spaces. In general, we study the asymptotic behavior and, in some particular cases, we can determine the sharp constants through the theory of reproducing kernel Hilbert spaces. As an application, we will present some connections with other extremal problems and a direct link with sharp Poincaré inequalities.

Rubio de Francia’s extrapolation theorem allows one to show that an operator that is bounded on weighted Lebesgue spaces for a single exponent and with respect to all weights in the associated Muckenhoupt class has to also be bounded for every exponent. As a matter of fact, in the previous years it has been shown that the operator has to be bounded on a much larger class of weighted spaces, including weighted Lorentz, variable Lebesgue, and Morrey spaces. All these spaces are examples of the generalized notion of Banach function spaces. In this talk I will discuss a unification and extension to a limited range setting of some of these results by presenting an extrapolation theorem in this general framework. I will also discuss an application to compact extrapolation that is part of a joint work with Emiel Lorist.