Mathematics Graduate Students Organization

Seminar:

Date: April 30

The study of singular integral operators (SIOs) and their kernels can be traced all the way back to Cauchy's work in complex analysis. In particular, mathematicians have been interested in the boundedness properties of SIOs. That is, for what Lp spaces is integration against a singular kernel a bounded linear operator? For much of its early history, the field was restricted to complex analytic methods, and so the question of boundedness of SIOs in higher dimensions remained open. Shortly after World War II, Alberto Calderón and Antoni Zygmund developed some techniques and theory that freed the study of SIOs from the limits of complex analysis. These techniques are now standard tools in harmonic analysis, nonlinear analysis, and PDEs. We will discuss some of these methods as well as what sorts of kernels they are applicable to.