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

Given $n \in \mathbb{N}_*$, 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 $\lambda > 0$, $$\lambda^{n + 1} \big\vert\big\{ x \in \mathbb{B}^{n + 1} : \vert D U (x)\vert > \lambda\big\}\big\vert \le \gamma \Big(\int_{\mathbb{S}^n}\int_{\mathbb{S}^n} \frac{\vert u (y) - u (z)\vert^{n + 1}}{\vert y - z\vert^{2 n}} \,\mathrm{d} y \,\mathrm{d} z \Bigr)\ ,$$ 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.