The goal of this research project is to continue out investigation on a class of $C^\infty$ functions whose derivatives satisfy quantitative size estimates. The estimates, called {global $L^q$ Gevrey estimates}, first arose in the work of Boggess and Raich \cite{BoRa13h} when they investigated how to capture a particular type of exponential decay through estimates on the Fourier transform. In this project, we propose to refine the notion of global $L^q$-Gevrey functions and include a discussion of the function theory as well as the relationship to Gevrey classes and known function spaces. Additionally, we present explicit examples of global $L^q$-Gevrey functions and ways to generate new global $L^q$-Gevrey functions from old ones. We also want to investigate some applications: The first is solving a Carleman-type problem for constructing functions whose derivatives are a given sequence of global $L^q$-Gevrey functions. The other two applications concern extensions of a given global $L^q$-Gevrey function: the first is constructing an almost analytic extension, and the second is building an approximate solution to a first-order complex vector field whose coefficients are global $L^q$-Gevrey functions. (AU)