Old and new results on Fourier multipliers
Abstract: In this talk I will give the definition of Fourier multiplier and I will provide some basic example and counterexamples. In particular I will show that the characteristic function of the sphere is not a Fourier multiplier. This is a celebrated result of C. Fefferman (1971). Then I will define the Cauchy transform on a cone and I will show some new result on the L^p boundness of this operator. I will also present a number of open problems and possible direction of research in this field.