SUBELLIPTIC PSEUDO-DIFFERENTIAL OPERATORS AND FOURIER INTEGRAL OPERATORS ON COMPACT LIE GROUPS

This memoir is devoted to the development of the theory of pseudo-differential operators with respect to a sub-Riemannian structure on an arbitrary compact Lie group. Our approach recovers the elliptic theory developed by the second author and V Turunen which in some sense is a theory of pseudo-differential operators adapted to the Laplacian on a compact Lie group. This means that the Hörmander classes are defined in terms of the spectrum of the Laplacian and they are compatible with the sharp analysis of elliptic problems. Indeed, in the latter case, the decay or growth of the symbols and their derivatives/differences is measured in terms of the spectrum of the Laplacian. However, in the setting of the subelliptic classes that we develop here, such a decay or growth is measured in terms of the spectrum of a Hörmander sub-Laplacian and then our theory allows the sharp analysis of subelliptic problems.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets