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Abstract
In this thesis, we study pseudodifferential operators on a realanalytic manifold, which is either compact Riemannian, or a Lie group with a biinvariant metric. Our aim is to obtain algebras of such operators, acting on realanalytic functions, but preserving a tube domain into which the functions extend holomorphically. The tube domain, contained in a complexification, is known as a ”Grauert tube”. We show that all the operators commuting with the Laplacian have this property, and in so doing, we make use of the Poisson transform introduced by Stenzel [56]. The transform is derived from a special case of a claim by Boutet de Monvel in [3], which was proved only recently by Stenzel [55] and Zelditch [65] in different ways. We demonstrate that the same would be true of many other realanalytic operators, if Boutet de Monvel’s claim holds in general, and briefly discuss approaches to it. Finally, in the setting of operators on a Lie group carrying a biinvariant metric, without using the transform, we obtain a nontrivial algebra with this property.
This algebra is determined by a subspace of the global matrixvalued symbols, which was introduced by Ruzhansky, Turunen and Wirth [49].
This algebra is determined by a subspace of the global matrixvalued symbols, which was introduced by Ruzhansky, Turunen and Wirth [49].
Original language  English 

Publisher  Technical University of Denmark 

Number of pages  138 
Publication status  Published  2021 
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 1 Finished

Pseudo  Differential operators for analytic boundary problems
Winterrose, D. S., Solovej, J. P., Stenzel, M. B., Gravesen, J. K., Karamehmedovic, M. & Brander, D.
Technical University of Denmark
01/06/2018 → 30/09/2021
Project: PhD