I don't understand this definition of the Sobolev space $H^{r,s}(E)$ for a vector bundle:
Def. The section $s:M \rightarrow E$ is contained in the Sobolev space $H^{r,s}(E)$ if for any bundle atlas with the property that on compact sets all coordinate changes and their derivatives are bounded, and for any bundle chart from that atlas, $\phi: E_U \rightarrow U \times R^n$, we have that $\phi \circ s_{|U}$ is contained in $H^{r,s}(U)$.
I understand that the requirement "on compact sets all coordinate changes and their derivatives are bounded" is for compatibility reasons but I don't know what does $H^{r,s}(U)$ mean.
Would the definition of $L^2$ section be the same except for the requirement on the derivatives to be bounded?
Thanks.