Addendum to “The equivariant spectral function of an invariant elliptic operator. Lp-bounds, caustics, and concentration of eigenfunctions” [J. Math. Pures Appl. 120 (2018) 33–90

Abstract

Let M be a compact boundaryless Riemannian manifold, carrying an effective and isometric action of a torus T, and P0 an invariant elliptic classical pseudodifferential operator on M. In this note, we strengthen the asymptotics for the equivariant (or reduced) spectral function of P0 derived in [6], which are already sharp in the eigenvalue aspect, to become almost sharp in the isotypic aspect. In particular, this leads to hybrid equivariant Lp-bounds for eigenfunctions that are almost sharp in the eigenvalue and isotypic aspect.