On regularization of vector distributions on manifolds

TL;DR

This paper develops a topological framework for representing vector-valued distributions on manifolds and characterizes continuous linear operators between them using tensor products and category theory.

Contribution

It introduces a topological isomorphism framework for vector-valued distributions and operators on manifolds, utilizing category theory and Schwartz's epsilon-product.

Findings

Established isomorphisms in the category of locally convex modules.

Represented operators as sections of tensor products with distributional coefficients.

Unified the theory using category-theoretic formalism and epsilon-products.

Abstract

One can represent Schwartz distributions with values in a vector bundle $E$ by smooth sections of $E$ with distributional coefficients. Moreover, any linear continuous operator which maps $E$-valued distributions to smooth sections of another vector bundle $F$ can be represented by sections of the external tensor product $E^* \boxtimes F$ with coefficients in the space $\mathcal{L}(\mathcal{D}', C^\infty)$ of operators from scalar distributions to scalar smooth functions. We establish these isomorphisms topologically, i.e., in the category of locally convex modules, using category theoretic formalism in conjunction with L. Schwartz' notion of $\varepsilon$-product.