Singular integral operators and boundary problems on weighted Morrey spaces and their pre-duals.

This dissertation is an expanded version of the monograph Weighted Morrey Spaces: Singular Integrals and Boundary Problems, which is scheduled to appear in DeGruyter's Studies in Mathematics series, by Marcus Laurel and Marius Mitrea [74]. The principal aim of this thesis is to solve a variety of boundary value problems for weakly elliptic systems in rough domains with boundary datum in Muckenhoupt-weighted Morrey spaces as well as their pre-duals. This is a multifaceted problem which requires tools from a myriad of mathematical disciplines including geometric measure theory, harmonic analysis, and functional analysis. We build a functional analytic framework for Muckenhoupt-weighted Morrey spaces on Ahlfors regular sets and notably prove an extrapolation result involving this scale of spaces. This leads to the development of a Calderón-Zygmund theory involving this brand of Morrey space. In addition to identifying weighted Block spaces as the pre-dual of a weighted Morrey space, we also build from the ground up a Calderón-Zygmund theory for these Block spaces. An important feature which goes beyond the standard Calderón-Zygmund theory is small-norm estimates involving the BMO seminorm of the outward unit normal for Calderón-Zygmund operators that have an integral kernel with a chord-dot-normal structure. Such small-norm estimates become relevant as we use the method of layer potentials to solve our boundary problems. The crux of this method relies on solving an integral equation, which can be done by inverting a particular operator via a Neumann series, something that requires a sufficiently small operator norm for the boundary-to-boundary double layer. We obtain well-posedness results for the Dirichlet, Neumann, (homogeneous and inhomogeneous) Regularity Problem, and the Transmission Problem in the class of δ-AR flat domains for weakly elliptic, second-order, homogeneous, constant coefficient systems that can be written in a way so that their double layer potential has a chord-dot-normal structure.