Global gauges and global extensions in optimal spaces

We consider the problem of extending functions \phi:\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.