Lipschitz Spaces on Compact Manifolds

Let f be a bounded function on the real line IF!. One may characterize the structural properties off by the modulus of smoothness w(t,f) = sup{lf (4 -f( y)l; x, y E 08, I x y I < t>, and, if w(t) is a continuous nondecreasing function of t > 0 such that w(O) = 0, by the Lipschitz class Lip(w) which is the set of all continuous functions such that su~~<~ and Li pb) is a well defined subspace of C(M), the space of continuous functions on M. (For definitions and properties concerning manifolds we refer to Helgason [91)* The aim of this paper is to define and characterize higher order Lipschitz spaces on compact manifolds. There are several ways to introduce such concepts. For instance, if M is a Lie group with unit element e, having a right and left invariant Riemannian metric p, one may proceed as on the real line and define the r-th modulus of smoothness of the bounded function f by