Uniform boundedness principles for Sobolev maps into manifolds

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.

28 pages

References in corpus (3)

• A really simple elementary proof of the uniform boundedness theorem
• Higher order weak differentiability and Sobolev spaces between manifolds
• Weak approximation by bounded Sobolev maps with values into complete manifolds

Cited by in corpus (3)

• Trace theory for Sobolev mappings into a manifold
• Minimal $W^{s,\frac{n}{s}}$-harmonic maps in homotopy classes
• Quantitative characterization of traces of Sobolev maps