Uniform boundedness principles for Sobolev maps into manifolds

TL;DR

This paper extends the classical Banach--Steinhaus uniform boundedness principle to nonlinear Sobolev spaces of maps into manifolds, establishing that qualitative properties imply uniform quantitative bounds for various problems.

Contribution

It introduces a nonlinear boundedness principle for Sobolev maps into manifolds, linking qualitative properties to uniform bounds in trace, approximation, lifting, and superposition problems.

Findings

Qualitative properties imply uniform bounds in Sobolev map problems.

Results apply to trace surjectivity, weak approximation, lifting, and superposition.

Establishes a nonlinear analogue of the Banach--Steinhaus principle.

Abstract

Given a connected Riemannian manifold $\mathcal{N}$, an \(m\)--dimensional Riemannian manifold $\mathcal{M}$ which is either compact or the Euclidean space, $p\in [1, +\infty)$ and $s\in (0,1]$, we establish, for the problems of surjectivity of the trace, of weak-bounded approximation, of lifting and of superposition, that qualitative properties satisfied by every map in a nonlinear Sobolev space $W^{s,p}(\mathcal{M}, \mathcal{N})$ imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach--Steinhaus uniform boundedness principle in linear Banach spaces.