Sharp estimate of global Coulomb gauge

Abstract

Let $A$ be a $W^{1,2}$-connection on a principle $\text{SU}(2)$-bundle $P$ over a compact $4$-manifold $M$ whose curvature $F_A$ satisfies $\Vert F_A{\Vert }_{L^2(M)}\le \Lambda$. Our main result is the existence of a global section $\sigma :M\to P$ with finite singularities on $M$ such that the connection form ${\sigma }^{\ast }A$ satisfies the Coulomb equation $d^{\ast }({\sigma }^{\ast }A)=0$ and admits a sharp estimate $\Vert {\sigma }^{\ast }A{\Vert }_{{\mathcal{L}}^{4,\infty }(M)}\le C(M,\Lambda )$. Here ${\mathcal{L}}^{4,\infty }$ is a new function space we introduce in this paper that satisfies $L^4(M)⊊{\mathcal{L}}^{4,\infty }(M)⊊L^{4-\epsilon }(M)$ for all $\epsilon >0$.