Sobolev spaces of integer order on compact homogeneous manifolds and invariant differential operators (Q1357130)

A description of the Sobolev-space \(W^1_p(M)\) (with \(1<p<\infty\)) for homogeneous compact oriented manifolds \(M=G/K\) is given in terms of the translation condition \[ f\in W^1_p(M) \Leftrightarrow \sup_{0\neq X\in\mathfrak g} \frac{\|f{\o}\tau_{\exp X}-f\|_{L^p(M)}}{|X|}<\infty, \] where \(\mathfrak g\) denotes the Lie-algebra of \(G\) and \(\tau\) the transitive action of \(G\) on \(M\). This is used to give a regularity result for solutions of the PDE \(u-Lu=f\) on \(M\), where \(L\) is a second-order differential operator generalizing the Laplace-Beltrami operator.