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?