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

Given $n\in {\mathbb{N}}_{\ast }$, a compact Riemannian manifold $M$ and a Sobolev map $u\in W^{n\mathrm{/}(n+1),n+1}({\mathbb{S}}^n;M)$, we construct a map $U$ in the Sobolev-Marcinkiewicz (or Lorentz-Sobolev) space $W^{1, (n + 1, \infty)} ($\mathbb{B}$^{n + 1}; M)$suchthat$u = U$inthesenseoftraceson$$\mathbb{S}$^{n} = \partial $\mathbb{B}$^{n + 1}$ and whose derivative is controlled: for every $\lambda >0$, $${\lambda }^{n+1}\mid \{x\in {\mathbb{B}}^{n+1}:\mathrm{\mid }DU(x)\mathrm{\mid }>\lambda \}\mid \le \gamma \left({\int }_{{\mathbb{S}}^n}{\int }_{{\mathbb{S}}^n}\frac{\mathrm{\mid }u(y)-u(z){\mathrm{\mid }}^{n+1}}{\mathrm{\mid }y-z{\mathrm{\mid }}^{2n}}\mathrm{d}y\mathrm{d}z\right),$$ where the function $\gamma :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ 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.