Joint BCAM-UPV/EHU Analysis and PDE seminar: Singular integrals: Multi-parameter framework and Zygmund dilation

Over a half-century ago, Alberto Calderón and Antoni Zygmund made groundbreaking studies that initiated the study of modern harmonic analysis that focuses on singular integrals. These operators still lie at the core of today's harmonic analysis. Recently, studies of singular integrals have been fast-paced due to new techniques that are based on similar underlying methods as in the classical studies by Calderón and Zygmund. These new techniques, namely the sparse method and the representation theory, have enabled us to find, for example, sharp weighted estimates and handle more complex operators such as multi-parameter singular integrals. Multi-parameter framework refers to the study of singular integrals whose singularity is more delicate to work with as it is expanded over the underlying space.

In this talk, I will be giving an introduction to the study of singular integrals and extensions to the multi-parameter framework, where we will be focusing, especially on the kernels that are dilation invariant under a specific "entangled" dilation - the so-called Zygmund dilation.