Unimodular Fourier multipliers

María Jesús Carro, Universidad Complutense de Madrid

In this talk, we will focus on the theory of Fourier multipliers on Lebesgue spaces. We will begin by reviewing some important results in the theory, which will motivate the following open problem: characterizing those functions m that are Fourier multipliers on Lp for p ≠ 1, 2.

Among these multipliers, we will consider those satisfying the additional condition that their modulus is identically one. Such multipliers are called unimodular Fourier multipliers. We will show that, in order to understand the set of all Fourier multipliers, it is enough to study the unimodular ones.

The proof of our main theorem relies on a novel extension of Stein’s interpolation theorem for analytic families of operators. Since we will try to avoid technicalities, we will also review some of the basic ideas and tools from interpolation theory that are relevant to the proof.