Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the completition of $C^\infty(X, E)$ with the norm
$$ \lVert f \rVert_{W^{k,p}} = \int_X |f|^p + |\nabla_{A_0}f|^p + \cdots + |\nabla^k_{A_0} f|^p ) d\mathrm{vol}. $$ My question is about Sobolev multiplication: If we have two elements $f\in W^{1,2}(E)$ and $g\in W^{1,2}(F)$, when is it true that $f\otimes g\in W^{1,2}(E\otimes F)$?.