On the $L^p$ index of spin Dirac operators on conical manifolds
Volume 177 / 2006
Studia Mathematica 177 (2006), 97-112
MSC: Primary 58J20.
DOI: 10.4064/sm177-2-1
Abstract
We compute the index of the Dirac operator on a spin Riemannian manifold with conical singularities, acting from $L^p({\mit\Sigma} ^+)$ to $L^q({\mit\Sigma} ^-)$ 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 $1+{n}/{p}-{n}/{q}\leq 0$ the index formula contains an extra term related to the Calderón projector.
