María Jesus Carro, Universidad Complutense de Madrid

In this talk, we will discuss Fourier multipliers on Lebesgue spaces. We will start by reviewing some important classical results and introducing a central open problem: how to characterize Fourier multipliers on L^p when (p \neq 1,2). We will then focus on a special class of multipliers, called unimodular Fourier multipliers, whose modulus is always one. We will show that studying these multipliers is enough to understand all Fourier multipliers. Our main result is based on a new extension of Stein’s interpolation theorem for analytic families of operators. To make the talk accessible, we will avoid technical details and introduce the basic ideas from interpolation theory needed to understand the idea of the proof.