Uniform boundedness principles for Sobolev maps into manifolds

Given a connected Riemannian manifold $\mathcal{N}$, an $m$–dimensional Riemannian manifold $\mathcal{M}$ which is either compact or the Euclidean space, $p\in [1,+\mathrm{\infty })$ 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.