Final Report Summary - IFIOP (Inequalities on Function Spaces and Properties of Integral Operators with Applications)
This project involved the study of integral operators, mathematical objects devised originally to help in the solution of differential equations, but subsequently studied for their intrinsic mathematical interest. The properties we investigated include boundedness — roughly, by how large a factor these operators scale up or down the data that it fed into them — and the closely related ideas of compactness and approximation numbers — these tell us how accurately these rather complex objects can be described by finite tables of numbers called matrices, which can be processed by computers. As well as the operators, we have to consider the function spaces on which they act. These are, essentially, containers for the data that is fed into the operator and the transformed data that it returns; our work mostly concerns the Lebesgue spaces, denoted Lp. Here p denotes a number; different values of p prove to be useful in different applications. We also work with the slightly more complicated Lorentz spaces Lp,q and Sobolev spaces Wp,s.