ICTS

Ultradifferentiable functions and ultradistributions are used in the general theory of linear partial differential operators such as in hypoellipticity, local solvability and propagation of singularities. In this talk, I will present a global characterisation of the ultradifferentiable functions, namely, Gevrey classes and the corresponding ul-tradistributions on compact Lie groups. Then I will discuss a characterisation of the Komatsu classes of ultradifferentiable functions and ultradistributions on compact manifolds. The above characterisations will be given in terms of the eigenfunction expansion of the Laplace-Beltrami operator on compact groups and an elliptic operator on the compact manifolds, respectively. The results on the compact manifolds extend the earlier characterisation of analytic functions on compact manifolds (by Seeley) and our results of Gevrey classes on compact Lie groups. The above analysis is based on the global Fourier analysis on compact Lie groups and manifolds. I will allude to the recent applications (by Ruzhansky et. al.) of the above results to show the well-posedness of Cauchy problems associated with the sums of squares of vector fields.