Controlled singular extension of critical trace Sobolev maps from spheres to compact manifolds

Abstract

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 $\lambda >0$, $${\lambda }^{n+1}\left|\right\{x\in {\mathbb{B}}^{n+1}:|DU(x)|>\lambda \left\}\right|\le \gamma \left({\int }_{{\mathbb{S}}^n}{\int }_{{\mathbb{S}}^n}\frac{|u(y)-u(z){|}^{n+1}}{|y-z{|}^{2n}}\mathrm{d}y\mathrm{d}z\right),$$ where the function $\gamma :[0,\infty )\to [0,\infty )$ 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.

26 pages, 1 figure

References in corpus (1)

• Global gauges and global extensions in optimal spaces

Cited by in corpus (1)

• Universality of renormalisable mappings in two dimensions: the case of polar convex integrands