DIFFERENTIAL OPERATORS ON SCHWARTZ DISTRIBUTIONS

Abstract

Differential operators on Schwartz distributions usually are defined as the transpose of differential operators on test functions. However, they do not exhaust all differential operators. In a general setting, Schwartz distributions on sections of compact support of a vector bundle Y over a smooth manifold X are considered. They constitute a C∞(X)-module. We follow a generic algebraic notion of differential operators on a module over a commutative ℝ-ring C∞(X). Such a differential operator on Schwartz distributions is proved to be the transpose of a differential operator on sections of Y → X if and only if it is continuous.