Global gauges and global extensions in optimal spaces

Abstract

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