Compact Sobolev Embeddings and the Equivariant Spectral Function of an Invariant Elliptic Operator with Full Asymptotics of Analytic Torsion

Abstract

The Hardy inequality, compact Sobolev embeddings and L^p-estimates for the torsion functions are studied. We find the estimates for the torsion functions with Robin or Dirichlet boundary conditions and Sobolev constants. We determine the L^p norm of the spectral clusters and of higher rank eigenfunctions for compact manifolds with boundary and bounds for spherical functions. The equivariant spectral function of a Riemannian orbifold, invariant elliptic operator, L^p-bounds, caustics and concentration of eigenfunctions are obtained. We show the holomorphic and the asymptotics of the analytic torsion on Hermitian symmetric spaces and on CR manifolds with S^1 domain and the orbifold submersion with the full asymptotics of analytic torsion.