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This is taken from Jost's text:

We let $E$ be a vector bundle over $M$, $s : M \to E$ be a section of $E$ with compact support. We say that $s$ is contained in the Sobolev space $H^{k,r}(E)$, if for any bundle atlas with the property that on compact sets all coordinates changes and all their derivatives are bounded, and for any bundle chart from such an atlas, $$\varphi : E\vert_U \to U \times \mathbb{R}^n$$ we have that $\varphi \circ s\vert_U$ is contained in $H^{k,r}(U)$.

Two questions:

  • Why do we require that the coordinate changes and all their derivatives are bounded? How does that have anything to do with $s$ being in the Sobolev space?
  • By the definition of $\varphi$, wouldn't we have to require that $\varphi \circ s\vert_U \in H^{k,r}(U \times \mathbb{R}^n)$?

1 Answers1

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  1. If you don't require bounds on the coordinate change maps then this definition becomes coordinate-dependent: you could have two charts $\varphi_1, \varphi_2$ disagreeing on whether or not $s$ is Sobolev. The assumption given is stronger than strictly necessary - it should suffice to just require that the first $k$ derivatives are bounded.

  2. Remember that $s|_U : U \to E|_U$ and $\varphi : E|_U \to U \times \mathbb R^n$ both respect fibres; i.e. $s(x) \in E_x$ and $\varphi(v) \in \pi_E(v) \times \mathbb R^n$. Thus the composition $\varphi \circ s|_U$ is a map $U \to U \times \mathbb R^n$ such that $\varphi \circ s(x) \in x\times \mathbb R^n;$ so it is the graph of some function $U \to \mathbb R^n$. It is the components $U \to \mathbb R$ of this function that we require to be in $H^{k,r}(U)$.

Anthony Carapetis
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