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Controlled singular extension of critical trace Sobolev maps from spheres to compact manifolds. (English) Zbl 1405.58002

Summary: Given \(n \in \mathbb N_\ast\), a compact Riemannian manifold \(M\) and a Sobolev map \(u \in W^{n/(n+1),n+1} (\mathbb S^n; M)\), we construct a map \(U\) in the Sobolev-Marcinkiewicz (or Lorentz-Sobolev) space \(W^{1,(n+1,\infty)} (\mathbb B^{n+1}; M)\) such that \(u=U\) in the sense of traces on \(\mathbb S^n=\partial \mathbb B^{n+1}\) and whose derivative is controlled: for every \(\lambda >0\), \[ \lambda^{n+1}| \{x \in \mathbb B^{n+1}:| DU(x)| > \lambda \}| \leq \gamma \left(\int_{\mathbb S^n} \int_{\mathbb S^n} \frac{| u(y) - u(z) |^{n+1}}{| y - z|^{2n}}\mathrm{d}y\, \mathrm{d}z\right), \] where the function \(\gamma :[0,\infty) \rightarrow [0,\infty)\) only depends on the dimension \(n\) and on the manifold \(M\). The construction of the map \(U\) relies on a smoothing process by hyperharmonic extension and radial extensions on a suitable covering by balls.

MSC:

58C25 Differentiable maps on manifolds
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