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

Mircea Petrache, Jean Van Schaftingen

Abstract:

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