The authors obtain global characterizations of the spaces of Gevrey ultradifferentiable functions or ultradistributions on a compact Lie group using the eigenvalues of the Laplace-Beltrami operator. Both the Roumieu and Beurling cases are considered. The paper also includes the corresponding characterizations on compact homogeneous spaces. To illustrate the type of obtained results we include below one of the main theorems. To this end some notation is needed.
Let \(X_1,\ldots, X_n\) be a basis of the Lie algebra of \(G\). For a multi-index \(\alpha = (\alpha_1,\ldots,\alpha_n)\) we denote by \(\partial^\alpha\) the composition of left-invariant derivatives with respect to \(X_1,\ldots,X_n\) such that each \(X_k\) enters exactly \(\alpha_k\) times. Then the Gevrey-Roumieu class \(\gamma_s(G)\) (\(s > 0\)) consists of those functions \(\phi\in C^\infty(G)\) for which there exist constants \(A > 0\) and \(C > 0\) such that \[ \sup_{x\in G}\left|\partial^\alpha\phi(x)\right|\leq CA^{|\alpha|}(\alpha!)^s. \] Theorem 2.3 (a) proves that \(\phi\in\gamma_s(G)\) if, and only if, there exist \(B > 0\) and \(K > 0\) such that \[ \parallel\widehat{\phi}(\xi)\parallel_{HS}\leq K e^{-B\left(1+\lambda^2_{[\xi]}\right)^{\frac{1}{2s}}} \] for all \(\xi\in\widehat{G}\). Here, \(\widehat{G}\) denote the set of (equivalent classes of) continuous irreducible unitary representations of G and \(\widehat{\phi}(\xi) = \int_G\phi(x)\xi(x)^\ast\;dx\), where \(dx\) is the normalized Haar measure of \(G\) and \(\xi:G\to {\mathbb C}^{d_\xi}\times {\mathbb C}^{d_\xi}\). The matrix elements of \(\xi\) are the eigenfunctions for the Laplace-Beltrami operator with eigenvalue \(-\lambda^2_{[\xi]}\).