Bilinear Littlewood-Paley Square Functions and Singular Integrals

Abstract

In this dissertation we further develop the bilinear theory of vector valued Calderoón-Zygmund operators, Littlewood-Paley square functions, and singu- lar integral operators. These areas of harmonic analysis are motivated by po- tential theory, boundary value problems in partial differential equations, har- monic and analytic extension problems in complex analysis, and many other classical problems in analysis. Multilinear operator theory addresses difficul- ties that arise from product type operations in harmonic analysis. We first introduce Banach valued Calderoón-Zygmund operators in a bilinear setting, and prove weak endpoint estimates and interpolation results for them. By viewing Littlewood-Paley square functions as Calderoón-Zygmund operators taking values in a particular Banach space, we are able to obtain bounds of the square functions on product Lebesgue spaces for a complete set of in- dices. We give an in depth analysis of Littlewood-Paley square functions, which includes estimates on some products of smooth function spaces as well as the estimates on product Lebesgue spaces that are needed to apply the vec- tor valued Calderoón-Zygmund results. Finally, we prove boundedness criteria for a certain class of bilinear singular integral operators on product Lebesgue spaces using Littlewood-Paley square function techniques. We provide a new proof of the bilinear T1 theorem that does not rely on the linear version of the result. We also prove a bilinear Tb theorem, a result missing in the theory so far. The Littlewood-Paley square function techniques developed in this work are a powerful tool has potential to solve problems in areas like oscillatory integral operator theory, multiparameter operator theory, Fourier restriction, and non-linear partial differential equations.