Uniform boundedness principles for Sobolev maps into manifolds. (English) Zbl 1418.46035
Summary: 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.
MSC:
| 46T20 | Continuous and differentiable maps in nonlinear functional analysis |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46T10 | Manifolds of mappings |
| 58D15 | Manifolds of mappings |
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