09-17 Notes

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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 a Sobolev norm by $\Vert \xi \Vert^p_{L^p_{j,A}} = \sum_{i=0}^j \int_X |\nabla_A^i\xi|^p d\mathrm{vol}$.