Smoothing operators for vector-valued functions and extension operators

TL;DR

This paper develops smoothing and extension operators for vector-valued functions on manifolds, providing new tools for approximating and extending functions and sections with applications to manifold mappings and fibre bundles.

Contribution

It introduces continuous linear smoothing operators and right inverses for restriction maps in various vector-valued function spaces on manifolds, extending existing theory.

Findings

Construction of smoothing operators S_n converging to the identity in C^k topology.

Existence of continuous linear right inverses for restriction maps.

Smoothing results for sections in fibre bundles.

Abstract

For suitable finite-dimensional smooth manifolds M (possibly with various kinds of boundary or corners), locally convex topological vector spaces F and non-negative integers k, we construct continuous linear operators S_n from the space of F-valued k times continuously differentiable functions on M to the corresponding space of smooth functions such that S_n(f) converges to f in C^k(M,F) as n tends to infinity, uniformly for f in compact subsets of C^k(M,F). We also study the existence of continuous linear right inverses for restriction maps from C^k(M,F) to C^k(L,F) if L is a closed subset of M, endowed with a C^k-manifold structure turning the inclusion map from L to M into a C^k-map. Moreover, we construct continuous linear right inverses for restriction operators between spaces of sections in vector bundles in many situations, and smooth local right inverses for restriction operators between manifolds of mappings. We also obtain smoothing results for sections in fibre bundles.