My primary research interests in harmonic analysis are the continuity (or boundedness) properties of multilinear and multiparameter integral operators on function spaces. These two types of operators are of fundamental importance to harmonic analysis and its applications, particularly in complex analysis and partial differential equations. The general goal of my research is to prove necessary and sufficient conditions for integral operators to be continuous between normed (or quasi-normed) function spaces, and apply them to boundary value problems related to analytic function.
The integral operators I focus on are multilinear and multiparameter Calderon-Zygmund singular integral operators and Littlewood-Paley square function operators. The typical form of my results are in what are colloquially referred to as T1 and Tb theorems. Generally speaking, a T1 theorem is a necessary and sufficient result that characterizes continuity properties of an operator T on function spaces by testing the action of T on the function 1 (hence the name T1 theorem). Its relative - the Tb theorem - gives an analogous characterization by testing the operator T on a certain type of perturbation function b in place of 1 (hence the name Tb theorem). In practice, the crux of these results is that one can conclude the continuity behavior of an operator T acting on an infinite collection of functions by only testing its behavior on the constant function 1, or in the case of the Tb theorem, on the function b. The function spaces on which I typically consider these operators are Lebesgue spaces, Hardy spaces, and Sobolev spaces. These continuity properties for Calderon-Zygmund and Littlewood-Paley operators are important in harmonic analysis for developing a complete theory to describe integral operators and their properties, which include aspects of frequency decompositions, properties of averaging operators, and many other principles in analysis. This theory in turn is a powerful analysis tool in many areas of mathematics.