*Published Paper*

**Inserted:** 21 sep 2024

**Journal:** J. Funct. Anal.

**Pages:** 21

**Year:** 2024

**Doi:** https://doi.org/10.1016/j.jfa.2024.110681

**Abstract:**

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 $W^{m/(m + 1), m + 1} (\mathbb{S}^m, \mathcal{N})$, a map $U : \mathbb{B}^{m + 1}_{1} \to \mathcal{N}$ 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.

**Keywords:**
Extension of traces in Sobolev spaces, trace theory, Sobolev embedding theorem, weak-type Marcinkiewicz spaces, Lorentz space