Microglobal regularity and the global wavefront set

Abstract

In this paper, we begin the study of regularity of partial differential equations in the space of global $$L^q$$ Gevrey functions, recently introduced in Adwan et al. (J Geom Anal 27(3):1874–1913, 2017) and Hoepfner and Raich (Indiana Univ Math J, forthcoming) and in a generalized and new function space called the space of global $$L^q$$ Denjoy–Carleman functions. We develop a wedge approach similar to Bony’s theorem (Bony in Seminaire Goulaouic–Schwartz (1976/1977), Equations aux derivees partielles et analyse fonctionnelle, Exp No 3. Centre Math, Ecole Polytech, Palaiseau, 1977) and prove three main theorems. The first establishes the existence of boundary values of continuous functions on a wedge. Next, we borrow the FBI transform approach from Hoepfner and Raich (forthcoming) to define global wavefront sets and prove a relationship between the inclusion of a direction in the global wavefront set and the existence of boundary values of sums of weighted $$L^p$$ functions defined in wedges. The final result is an application in which we prove a global version of a classical result: namely, the relationship between the global characteristic set of a partial differential operator P and the microglobal wavefront sets of u and Pu.