Optimal regularization processes on complete Riemannian manifolds

TL;DR

This paper introduces optimal regularization methods for Schwartz distributions on complete Riemannian manifolds, utilizing wave equation solutions to preserve microlocal structures and isometry invariance, enabling embeddings into generalized function algebras.

Contribution

It develops a novel regularization framework based on wave equation operators that maintains microlocal properties and symmetry, advancing distribution theory on manifolds.

Findings

Regularization preserves microlocal structure

Operators commute with isometries

Provides sheaf embeddings into generalized function algebras

Abstract

We study regularizations of Schwartz distributions on a complete Riemannian manifold $M$. These approximations are based on families of smoothing operators obtained from the solution operator to the wave equation on $M$ derived from the metric Laplacian. The resulting global regularization processes are optimal in the sense that they preserve the microlocal structure of distributions, commute with isometries and provide sheaf embeddings into algebras of generalized functions on $M$.