# Sobolev Multiplication theorem for Fibre bundles

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?

• If $M$ is just a manifold admitting an action, how are you defining a metric on the total space $E(M)$ for your embedding? Also, you seem to be treating $\Gamma(X, E(M))$ as if it is a vector space. – Paul Reynolds Sep 15 '15 at 16:31
• Thanks Paul for pointing it out! $M$ is a smooth, Riemannian manifold. I don't understand your second remark though. – Varun Sep 16 '15 at 4:41

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.
• Thanks a ton! That helps. For the case I study, I have that apriori, $u \in L^{1}_{2}\cap L^{\infty}$. From your answer, I gather that it won't suffice to obtain an $L^2$-bound on $\Theta\circ (u|_{U_{\alpha}})$. – Varun Sep 16 '15 at 6:35