Some principal characters of harmonic analysis are singular integrals, whose properties depend on delicate cancellation between quantities of opposite signs. In this project, we study a spectrum of such phenomena, which makes a cross section of harmonic analysis from one border to another. In one direction, there is the theory of abstract Banach spaces, which has been previously connected to various classical singular integrals. The aim of this project is to find similar connections and thus a new viewpoint to some recent achievements of harmonic analysis. In the other end, we have partial differential equations of many applications, and certain degenerate forms of them will be studied in this project via weighted norm inequalities for singular integrals. Between the two, there is a range of problems, of harmonic analysis proper and of the theory of function spaces, where the applicant's path-breaking dyadic representation methods, subsequently refined by others, play a key role. |