Topics on Singular Integral Operators over the Unit Disk(2/2)

There is a rich and sophisticated theory of singular integral operatorsover the whole Euclidean spaces in harmonic analysis. The theory hasdeep and far‐reaching applications to many areas including PDE. When itcomes to domains with boundary, current approach to singular integraloperators is usually via the theory of homogeneous spaces. Thisapproach results in complicated theorems, but seldom producesconcrete results for concrete operators. We feel that this approach is notsatisfactory, and we intend to build a program of singular integraloperators over the unit disk, the first example of a bounded domain.Specifically, we will develop three projects: (1) Bergman‐type singularintegral operators, (2) Riesz potential operators via Laplace equations, (3)general elliptic operators over bounded domains. A focus of our researchis super‐singularity, a new phenomenon which we observed recently. It isspecial to the unit disk, and is previous unknown for harmonic analysisover the whole space.