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 \(\left\|{\xi}\right\|^p_{L^p_{j,A}} = \sum_{i=0}^j \int_X |\nabla_A^i\xi|^p \ d\mathrm{vol}\). 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 \(A = A_0 + a\). We proceed by induction on \(j\). It will suffice to show that \(\left\|{\nabla_{A_0 + a}\xi}\right\|_{L^p_{j,A_0}} \leq \left\|{\xi}\right\|_{L^p_{j+1,A_0}}(1+\left\|{a}\right\|_{L^p_{k,A_0}}).\) The left hand side of this inequality is less than \(\left\|{\nabla_{A_0}\xi}\right\|_{L^p_{j,A_0}} + \left\|{a\xi}\right\|_{L^p_{j,A_0}} \leq \left\|{\xi}\right\|_{L^p_{j+1,A_0}} + \left\|{a}\right\|_{L^p_{k,A_0}}\left\|{\xi}\right\|_{L^p_{j+1,A_0}}.\) In this last inequality, we have applied the multiplication theorem to the second term, and this requires \((j-n/p) \leq (j+1 - n/p) + (k-n/p),\) which is the same as \((k+1)p>n\).

We conclude that the \(L^p_{k+1,A}\) norms are all equivalent for \(A \in \mathcal{A}^p_k\). Note that the condition \((k+1)p>n\) is also the condition for gauge group \(\mathcal{G}^p_{k+1} \subset L^p_{k+1}(X,\mathrm{End}(E))\) to consist of continuous sections and act continuously on \(\mathcal{A}^p_k\).