Singular extension of critical Sobolev mappings under an exponential weak-type estimate

Given $m\in \mathbb{N}\setminus \{0\}$ and a compact Riemannian manifold $\mathcal{N}$, we construct for every map $u$ in the critical Sobolev space $W^{m\mathrm{/}(m+1),m+1}({\mathbb{S}}^m,\mathcal{N})$, a map $U:{\mathbb{B}}^{m+1}\to \mathcal{N}$ whose trace is $u$ and which satisfies an exponential weak-type Sobolev estimate. The result and its proof carry on to the extension to a half-space of maps on its boundary hyperplane and to the extension to the hyperbolic space of maps on its boundary sphere at infinity.