On the Lp index of spin Dirac operators on conical manifolds

We compute the index of the Dirac operator on a spin Riemannian manifold with conical singularities, acting from L-p(Sigma(+)) to L-q(Sigma(-)) with p, q > 1. When 1 + n/p - n/q > 0 we obtain the usual Atiyah-Patodi-Singer formula, but with a spectral cut at (n + 1)/2 - n/q instead of 0 in the definition of the eta invariant. In particular we reprove Chou's formula for the L-2 index. For 1 + n/p - n/q <= 0 the index formula contains an extra term related to the Calderon projector.