*Preprint*

**Inserted:** 25 feb 2013

**Year:** 2013

**Links:**
arXiv:1302.5659

**Abstract:**

We consider the problem of extending functions $\phi:S^n \to S^n$ to functions $u:B^{n+1}\to S^n$ for $n=2,3$. We assume $\phi$ to belong to the critical space $W^{1,n}$ and we construct a $W^{1,(n+1,\infty)}$-controlled extension u. The Lorentz-Sobolev space $W^{1,(n+1,\infty)}$ is optimal for such controlled extension. Then we use such results to construct global controlled gauges for $L^4$-connections over trivial $SU(2)$-bundles in $4$ dimensions. This result is a global version of the local Sobolev control of connections obtained by K. Uhlenbeck.

**Keywords:**
global gauges, Sobolev maps, Lorentz spaces