Long time behavior of the half-wave trace and Weyl remainders

Abstract

Given a compact Riemannian manifold $(M,g)$, Chazarain, Hörmander, Duistermaat, and Guillemin study the half-wave trace $\operatorname{HWT}_{M,g}(τ) \in \mathscr{S}'(\mathbb{R}_τ)$. From the asymptotics of the half-wave trace as $τ\to 0$, Hörmander deduces the now standard remainder $\smash{O(σ^{d-1}) = O(λ^{d/2-1/2})}$ in Weyl's law, where $d=\dim M$. Given a dynamical assumption implying additional local regularity, Duistermaat and Guillemin improve this to $o(σ^{d-1})$. By examining the Tauberian step in the argument, we show how a quantitative version \[N(σ) = Z(σ) + O(σ^{d-1}\mathcal{R}(σ)^{-1/2})\] of the Duistermaat-Guillemin result follows under slightly stronger hypotheses, these implying that the $(d-1)$-fold regularized half-wave trace \[\langle D_τ\rangle^{1-d} \operatorname{HWT}_{M,g}(τ)\] is in $\smash{L^{1,1}_\mathrm{loc}(\mathbb{R}\backslash \{0\})}$. Here $Z(σ)\in \mathbb{R}[σ]$ is a polynomial and $\mathcal{R}(σ):\mathbb{R}^+\to \mathbb{R}^+$ is an $(M,g)$-dependent nondecreasing function with $\lim_{σ\to\infty} \mathcal{R}(σ)=\infty$, specified in terms of the growth rate of $\langle D_τ\rangle^{1-d} τ^{-1}\operatorname{HWT}_{M,g}(τ)$ as measured in $L^{1,1}$. Per Duistermaat-Guillemin, this hypothesis is implied by geometric conditions that hold ``generically'' for $d\geq 3$. Thus, we clarify the relation between the error term in Weyl's law and the long time behavior of the half-wave trace.