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.

To prove this, write $A = A_0 + a$. It will suffice to show that $\Vert \nabla_{A_0 + a}\xi \Vert_{L^p_{j,A_0}} \leq \Vert\xi\Vert_{L^p_{j+1,A_0}}(1+\Vert a\Vert_{L^p_{k,A_0}})$. The left hand side of this inequality is less than $\Vert \nabla_{A_0}\xi \Vert_{L^p_{j,A_0}} + \Vert a\xi \Vert_{L^p_{j,A_0}} \leq \Vert\xi\Vert_{L^p_{j+1,A_0} + \Vert a\Vert_{L^p_{k,A_0}}\Vert\xi\Vert_{L^p_{j+1,A_0}}$.