On Bojarski’s index formula for nonsmooth interfaces

Let $D$ be a Dirac type operator on a compact manifold ${\mathcal {M}}$ and let $\Sigma$ be a Lipschitz submanifold of codimension one partitioning ${\mathcal {M}}$ into two Lipschitz domains $\Omega _{\pm }$. Also, let ${\mathcal {H}}^{p}_{\pm }(\Sigma ,D)$ be the traces on $\Sigma$ of the ($L^{p}$-style) Hardy spaces associated with $D$ in $\Omega _{\pm }$. Then $({\mathcal {H}}^{p}_{-}(\Sigma ,D),{\mathcal {H}}^{p}_{+}(\Sigma ,D))$ is a Fredholm pair of subspaces for $L^{p}(\Sigma )$ (in Kato’s sense) whose index is the same as the index of the Dirac operator $D$ considered on the whole manifold ${\mathcal {M}}$.