Parameterizations and rectifiability via geometric functions and singular integral operators

Abstract

Geometric measure theory provides the framework to examine the geometry of sets and measures that are not ``smooth'' enough to be studied using the classical methods of differential geometry. These rough sets arise naturally in many settings, for instance as the minimizers of certain geometric variational problems, e.g \cite{R60}. A relatively recent technique used to analyze the fine properties of sets of this type is through the study of geometric square functions, which capture simple geometric information about the set at each scale. This technique has gained popularity starting with the work of Jones \cite{J90} and David-Semmes \cite{DS91}. In \cite{Bi87} a discontinuous geometric function appeared, known as the Carleson $\varepsilon$-function. In \cite{JTV21} and \cite{FTV23} the authors prove that qualitative control on the Carleson $\varepsilon$-function, and higher-dimensional analogues, characterize tangent points of certain domains. In this thesis, we study higher regularity versions of Carleson's conjecture in the plane and in higher dimensions. As for the study of the ``smoothness'' of a given measure on $\mathbb{R}^n$ a commonly used tool in geometric measure theory are \textit{singular integral operators}. The fine properties of measures involve two things: the ``smoothness'' of the set on which the measure lives and the behavior of the measure on that set. In \cite{DS91}, the authors prove that uniformly rectifiable measures are characterized by the $L^2$-boundedness of all Calder\'{o}n-Zygmund operators. In \cite{M95} and \cite{MP95}, the authors prove that the almost everywhere existence of principal values of the Riesz transform characterizes rectifiable measures. In this thesis we extend the work of \cite{M95, MP95} to a broader class of singular integral operators, in particular, a class of anisotropic Riesz kernels.