Commutators of Multilinear Singular Integral Operators with Pointwise Multiplication

Abstract

In this dissertation we further develop the theory of commutators of multilinear singular integral operators with pointwise multiplication. Generally speaking, the commutator of two operators is itself an operator that measures the changes which occur when switching the order in which the commuted operators are being applied. They have proven to be significant historically, and can be useful in the study of PDE. Our first main contribution is the completion of the characterization of the space functions with bounded mean oscillation (BMO) in terms of the boundedness of the corresponding commutator in an appropriate set of Lebesgue spaces. It is already known in a variety of settings that a function being in BMO is sufficient to conclude the boundedness of the commutator, we were able to show that this condition is in fact necessary, a long standing open question. Our characterization opened the door for us to obtain our second main result, namely the necessary and sufficient conditions which guarantee the compactness of the commutator of the bilinear singular integral with pointwise multiplication in appropriate weighted Lebesugue spaces.