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.
- Publication:
-
Annales de L'Institut Henri Poincare Section (C) Non Linear Analysis
- Pub Date:
- April 2019
- DOI:
- 10.1016/j.anihpc.2018.06.002
- arXiv:
- arXiv:1709.08565
- Bibcode:
- 2019AIHPC..36..417M
- Keywords:
-
- Mathematics - Functional Analysis;
- Mathematics - Analysis of PDEs;
- 46T20 (46E35;
- 46T10;
- 58D15)
- E-Print:
- 28 pages