Global $$L^q$$-Gevrey Functions and Their Applications

Abstract

The goal of this paper is to introduce a class of \(C^\infty \) functions derivatives of which satisfy quantitative size estimates. The estimates, called global \(L^q\) Gevrey estimates, first arose in the work of Boggess and Raich ( J. Fourier Anal. Appl. 19:180–224, 2013) when they investigated how to capture a particular type of exponential decay through estimates on the Fourier transform. In the present work, we 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. In addition, we present explicit examples of global \(L^q\)-Gevrey functions and ways to generate new global \(L^q\)-Gevrey functions from old ones. We conclude with three applications: The first is solving a Carleman-type problem for constructing functions derivatives of which 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 coefficients of which are global \(L^q\)-Gevrey functions.