Let $X$ be a compact, oriented, four dimensional Riemannian manifold and $Q\longrightarrow X$ be a principal $G$-bundle over $X$ for a smooth, compact Lie group $G$. Let $M$ be a smooth, Riemannian manifold admitting a left action of $G$. Denote, for simplicity, the associated bundle by $E(M):= Q\times\_{G}M$. Then $\Gamma(X, E(M))\cong C^{\infty}(Q,M)^{G}$, the latter being the space of smooth equivariant maps from $Q \longrightarrow M$. By Nash embedding theorem, there exists an $N\in \mathbb{N}$, such that the embedding $\iota:E(M)\hookrightarrow \mathbb{R}^N$ is an isometric embedding. We define the $L^{p}\_{k}$-norm of $u\in \Gamma(X, E(M))$ as $||u||\_{L^{p}\_{k}}:= ||\iota\circ u||\_{L^{p}\_{k}}$. We now define $L^{p}\_{k}(X,E(M))$ to be the completion of $\Gamma(X, E(M))$ in the $L^{p}\_{k}$-norm and for any $\hat{u} \in C^{\infty}(Q,M)^{G}$, $||\hat{u}||\_{L^{p}\_{k}} := ||u||\_{L^{p}\_{k}}$, where $u$ is the section associated to the map $u$. 


Let $u\in L^{p}\_{k}(Q,M)^{G}$ and $A\in\Omega^{1}(Q,\mathfrak{g})^{G}\_{L^{q}\_{l}}$ be an $L^{q}\_{l}$-connection on $Q$.
Define $A \cdot u := K^{M}\_{A}|\_{u}$, where $K^{M}\_{A}$ is the fundamental vector field on $M$ due to $A$ and along $u$, given by $K^{M}\_{A}|\_{u}(v) = K^{M}\_{A(v)}|\_{u}$. 


My question is, in what Sobolev space, does $K^{M}\_{A}|\_{u}$ belong? Does the Sobolev Multiplication Theorem
$L^{p}\_{k} \times L^{q}\_{l} \longrightarrow L^{r}\_{m},$ where,
$\frac{1}{r}-\frac{m}{4} > \frac{1}{p}-\frac{k}{4} + \frac{1}{q}- \frac{l}{4}$ (below borderline case) and $r = min(k,l)$, hold true in this setting? 



Sobolev Multiplication theorem for Fibre bundles
I have a solution in mind. But for that the embedding $E(M) \to \mathbb R^N$ is unnatural. So, I proceed in a slightly different way. Firstly, in any case it is required that $kp>4$, because the problem will involve left composing a Sobolev function with a smooth function. These operations are well-behaved only above the Sobolev borderline.
Making that assumption, $X$ has a cover $X=\cup\_\alpha U\_\alpha$, such that the bundle $Q|\_{U\_\alpha}$ is trivial; and after choosing a trivialization, $u(U\_\alpha)$ is contained in a chart of $M$. Then, locally we may assume $u$ is a map $u:U\_\alpha \to \mathbb R^m$ (where $m$ is the dimension of $M$).


The infinitesimal action of $\mathfrak g$ on $M$ induces a smooth section $\Theta:M \to \operatorname{End}(\mathfrak g,TM)$. Locally, this is a smooth section $\Theta:V \to \operatorname{End}(\mathfrak g,\mathbb R^m)$, where $V$ is a chart of $M$. By composition of functions, we get $\Theta \circ (u|\_{U\_\alpha}):U\_\alpha \to \operatorname{End}(\mathfrak g,\mathbb R^m)$ is in $L^p\_k$. Note that for this step, you either need $kp>4$ or that $u \in L^p\_k \cap C^0$. Now, $A.u$ is the multiplication of sections of vector bundles, namely it is $(\Theta \circ u)\cdot A$, for which relevant results in Sobolev multiplication can be used. 


