Controlled Singular Extension of Critical Trace Sobolev Maps from Spheres to Compact Manifolds

Petrache, Mircea [Max Planck Institute for Mathematics in the Sciences, Leipzig] Van Schaftingen, Jean [UCL]

Given n∈N∗, a compact Riemannian manifold M and a Sobolev map u∈W^(n/(n+1),n+1)(S^n;M), we construct a map U in the Sobolev-Marcinkiewicz (or Lorentz-Sobolev) space W^(1,(n+1,∞))(B^(n+1);M) such that u=U in the sense of traces on S^n=∂B^(n+1) and whose derivative is controlled: for every λ>0, λ^(n+1) ∣{x∈B^(n+1):|DU(x)|>λ}∣ ≤ γ(∫Sn∫Sn|u(y)−u(z)|^(n+1)/|y−z|^(2n) dy dz) , where the function γ:[0,∞)→[0,∞) 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.