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Global ultradifferentiable functions, complex analysis, and PDE's

Grant number: 18/02663-0
Support type:Research Grants - Visiting Researcher Grant - International
Duration: September 01, 2018 - October 31, 2018
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Analysis
Principal researcher:Gustavo Hoepfner
Grantee:Gustavo Hoepfner
Visiting researcher: Andrew Seth Raich
Visiting researcher institution: University of Arkansas, United States
Home Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil


We want to study regularity of partial differential equations inthe space of global $L^q$ Gevrey functions, recently introduced in [Z. Adwan, G. Hoepfner, and A. Raich, Global Lq-Gevrey functions and their applications, J. Geom. Anal. 27 (2017), no. 3, 1874-1913.] and [G. Hoepfner and A. Raich, Global Lq Gevrey functions, Paley-Weiner theorems, and the FBI transform, to appear, Indiana Univ. Math. J.] and in a generalized and new function space called the space of global $L^q$ Denjoy-Carleman functions.We aim to develop a wedge approach similar to a celebrated Bony's theorem [J. M. Bony, Équivalence des diverses notions de spectre singulier analytique, Séminaire Goulaouic-Schwartz (1976/1977), E quations aux dérivées partielles et analysefonctionnelle, Exp.No.3, CentreMath., École Polytech., Palaiseau,1977.] and among other results, we will focus on three main topics. The first establishes the existence of boundary values of continuous functions on a wedge. Next, we borrow the FBI transform approach from [G. Hoepfner and A. Raich, Global Lq Gevrey functions, Paley-Weiner theorems, and the FBI transform, to appear, Indiana Univ. Math. J.] to define global wavefront sets and try to 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 last is an classical application namely, the relationship between the global characteristic set of a partial differential operator $P$ and the microglobal wavefront sets of $u$ and $Pu$. We also want to investigate other relations with this $L^q$ Global Denjoy-Carleman functions such as: Hardy spaces properties and the Grusin operator. (AU)

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Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
HOEPFNER, GUSTAVO; RAICH, ANDREW. Microglobal regularity and the global wavefront set. MATHEMATISCHE ZEITSCHRIFT, v. 291, n. 3-4, p. 971-998, . (17/06993-2, 17/03825-1, 18/02663-0)
COACALLE, JOEL; RAICH, ANDREW. Closed Range Estimates for (partial derivative)over-bar(b) on CR Manifolds of Hypersurface Type. JOURNAL OF GEOMETRIC ANALYSIS, v. 31, n. 1, p. 366-394, . (18/02663-0)

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