Addendum to 'The equivariant spectral function of an invariant elliptic operator'

TL;DR

This paper improves asymptotic estimates for the equivariant spectral function of invariant elliptic operators on manifolds with torus actions, leading to nearly optimal bounds for eigenfunctions in both eigenvalue and isotypic aspects.

Contribution

It strengthens previous asymptotic results for the equivariant spectral function, achieving near-sharpness in the isotypic aspect and deriving hybrid $L^p$ bounds for eigenfunctions.

Findings

Asymptotic estimates for the spectral function are improved.

Hybrid $L^p$ bounds for eigenfunctions are established.

Results are nearly sharp in both eigenvalue and isotypic aspects.

Abstract

Let $M$ be a compact boundaryless Riemannian manifold, carrying an effective and isometric action of a torus $T$, and $P_0$ an invariant elliptic classical pseudodifferential operator on $M$. In this note, we strengthen asymptotics for the equivariant (or reduced) spectral function of $P_0$ derived previously, which are already sharp in the eigenvalue aspect, to become almost sharp in the isotypic aspect. In particular, this leads to hybrid equivariant $L^p$-bounds for eigenfunctions that are almost sharp in the eigenvalue and isotypic aspect.