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

Given a compact Riemannian manifold $(M,g)$, Chazarain, H\"ormander, Duistermaat, and Guillemin study the half-wave trace $\operatorname{HWT}_{M,g}(\tau) \in \mathscr{S}'(\mathbb{R}_\tau)$. From the asymptotics of the half-wave trace as $\tau\to 0$, H\"ormander deduces the now standard remainder $\smash{O(\sigma^{d-1}) = O(\lambda^{d/2-1/2})}$ in Weyl's law, where $d=\dim M$. Given a dynamical assumption implying additional local regularity, Duistermaat and Guillemin improve this to $o(\sigma^{d-1})$. By examining the Tauberian step in the argument, we show how a quantitative version $$N(\sigma) = Z(\sigma) + O(\sigma^{d-1}\mathcal{R}(\sigma)^{-1/2})$$ of the Duistermaat-Guillemin result follows under slightly stronger hypotheses, these implying that the $(d-1)$-fold regularized half-wave trace $$\langle D_\tau \rangle^{1-d} \operatorname{HWT}_{M,g}(\tau)$$ is in $\smash{L^{1,1}_\mathrm{loc}(\mathbb{R}\backslash \{0\})}$. Here $Z(\sigma)\in \mathbb{R}[\sigma]$ is a polynomial and $\mathcal{R}(\sigma):\mathbb{R}^+\to \mathbb{R}^+$ is an $(M,g)$-dependent nondecreasing function with $\lim_{\sigma\to\infty} \mathcal{R}(\sigma)=\infty$, specified in terms of the growth rate of $\langle D_\tau \rangle^{1-d} \tau^{-1}\operatorname{HWT}_{M,g}(\tau)$ 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.