Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

Michael Ruzhansky,Aparajita Dasgupta

doi:10.1016/j.bulsci.2013.12.001

Michael Ruzhansky, Aparajita Dasgupta

Open Access

https://doi.org/10.1016/j.bulsci.2013.12.001

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Abstract

In this paper we give global characterisations of Gevrey–Roumieu and Gevrey–Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace–Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their α-duals in the sense of Köthe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.

Highlights

The spaces of Gevrey ultradifferentiable functions are well-known on Rn and their characterisations exist on both the space-side and the Fourier transform side, leading to numerous applications in different areas
Sci. math. 138 (2014) 756–782 the spaces of ultradistributions using the eigenvalues of the Laplace–Beltrami operator LG (e.g. Casimir element) on the compact Lie group G
We will first give characterisations of the so-called α-duals of the Gevrey spaces and show that α-duals and topological duals coincide. We prove that both types of Gevrey spaces are perfect spaces, i.e. the α-dual of its α-dual is the original space

Summary

Introduction

The spaces of Gevrey ultradifferentiable functions are well-known on Rn and their characterisations exist on both the space-side and the Fourier transform side, leading to numerous applications in different areas. In terms of the group Fourier transform Eq (1.1) is basically with constant coefficients, and the global characterisation of Gevrey spaces together with an energy inequality for (1.1) yield the well-posedness result. We will address this and other applications elsewhere, but we note that in these problems both types of Gevrey spaces appear naturally, see e.g. If we want to emphasise the change of the constant, we may use letters like C , A1, etc

Results

For q

From and we obtain inf

For a given