Convolution of valuations on manifolds

TL;DR

This paper introduces a new convolution operation for valuations on manifolds with group actions, extending previous convolution concepts and establishing module structures over valuation algebras.

Contribution

It defines a novel convolution of valuations on manifolds with group actions and proves module properties, extending known convolution frameworks.

Findings

Established convolution as a module action on valuations

Derived explicit formulas for vector space cases

Extended previous convolution concepts by Fu and others

Abstract

We introduce the new notion of convolution of a (smooth or generalized) valuation on a group $G$ and a valuation on a manifold $M$ acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and generalized valuations on $M$ are modules over the algebra of compactly supported generalized valuations on $G$ satisfying some technical condition of tameness. The case of a vector space acting on itself is studied in detail. We prove explicit formulas in this case and show that the new convolution is an extension of the convolution on smooth translation invariant valuations introduced by J.~Fu and the second named author.