Remarks on global hypoellipticity

Abstract

We study differential operators <italic>D</italic> which commute with a fixed normal elliptic operator <italic>E</italic> on a compact manifold <italic>M</italic> . We use eigenfunction expansions relative to <italic>E</italic> to obtain simple conditions giving global hypoellipticity. These conditions are equivalent to <italic>D</italic> having parametrices in certain spaces of functions or distributions. An example is given by <italic>M</italic> = compact Lie group and and <italic>E</italic> = Casimir operator, with <italic>D</italic> any invariant differential operator. The connections with global subelliptic estimates are investigated.