Weighted distribution spaces and pseudodifferential operators

Abstract

There is a close relationship among classical spaces of functions and distributions, pseudodifferential operators of classical type, and the regularity theory of elliptic operators. Scales of spaces such as the standard Sobolev or H/51der spaces indexed by the degree of smoothness are mapped among themselves by standard pseudodifferential operators. This is true in particular of the (approximate) inverses of elliptic operators, a fact which is equivalent to the classical regularity theory. A similar relationship exists among certain spaces defined by non-isotropic quasihomogeneous smoothness conditions, the corresponding types of parabolic or other semi-elliptic operators, and the corresponding regularity theory. It is the purpose of this paper to develop an analogous theory for the spaces of functions and distributions naturally associated to general classes of pseudodifferential operators, with applications to the regularity theory of hypoelliptic operators.