09-17 Notes

Sobolev norms on sections

Let $A$ be a connection in a vector bundle $E \to X$. For $\xi$ a section of $E$, we may define the Sobolev $L^p_{j,A}$ norm by $\Vert \xi \Vert^p_{L^p_{j,A}} = \sum_{i=0}^j \int_X |\nabla_A^i\xi|^p \ d\mathrm{vol}$. The topology induced by this norm is independent of the connection $A$ in the following precise sense: If $A_0$ is a $C^\infty$ reference connection, and $A \in \mathcal{A}^p_k$ is any $L^p_k$ connection, then $L^p_{j,A}$ and $L^p_{j,A_0}$ norms are equivalent as long as the inequalities $(k+1)p > n$ and $0\leq j \leq k+1$ hold.