Solvability of noninvariant differential operators on homogeneous spaces

Abstract

We consider the solvability of non-invariant differential operators on homogeneous spaces. Such operators cannot be expected to have solutions in smooth functions (an illustrative example is provided). However, Lion has shown that, under suitable growth conditions on the infinitesimal components of the operators in a representation-theoretic decomposition, one can deduce solvability in a space of distributions. In this paper we prove that Lion’s result can be improved to yield solvability in square-integrable functions.