  The group of diffeomorphisms commuting with an elliptic operator on a
manifold is a compact Lie group under Compact-Open topology. In foliation
theory, pseudogroup is introduced by Sacksteder. The pseudogroup of local
transformations commuting with a basic differential operator possesses the
equicontinuity and the quasi-analyticity when conditions on operator are given.
These properties serve to construct a transverse metric on the normal bundle
under a good condition on operator. For this, the Average Method is applied as
in the construction of basic connection on foliated bundles.
