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Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 971–998 | Cite as

Microglobal regularity and the global wavefront set

  • Gustavo HoepfnerEmail author
  • Andrew Raich
Article
  • 37 Downloads

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 Séminaire Goulaouic–Schwartz (1976/1977), Équations aux dérivées partielles et analyse fonctionnelle, Exp No 3. Centre Math, École 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.

Keywords

FBI transform Wavefront set Global wavefront set Gevrey functions Global \(L^q\)-Gevrey functions Denjoy–Carleman functions Global \(L^q\) Denjoy–Carleman functions Ultradifferentiable functions Ultradistributions Almost analytic extensions 

Mathematics Subject Classification

42B10 35A18 26A12 26E10 58J15 42B37 42A38 35A27 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de São CarlosSão CarlosBrazil
  2. 2.Department of Mathematical Sciences, SCEN 3091 University of ArkansasFayettevilleUSA

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