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

Given $m\in N\setminus \left\{0\right\}$ and a compact Riemannian manifold $N$, we construct for every map u in the critical Sobolev space $W^{m/(m+1),m+1}(S^m,N)$, a map $U:B_1^{m+1}\to N$ 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.