Spectral cluster bounds for orthonormal functions on manifolds with nonsmooth metrics

TL;DR

The paper extends L^q spectral cluster bounds to families of orthonormal functions on compact manifolds with nonsmooth metrics. By adapting the spectral window width to the metric regularity using $\lambda^{(1-s)_+}$ and defining $\Pi_{\lambda}$ accordingly, it generalizes known smooth-metric results to $s\in[0,2]$, including precise exponents $\alpha(q)$ and $\delta_s(q)$ and localized cube bounds. In the $s=1$ (Lipschitz) and $s\in[1,2)$ regimes, it proves Franks–Sabin–type bounds for orthonormal systems with explicit dependence on the number of functions through $\alpha(q)$, $\delta_s(q)$, and a localized spatial scale $R=\lambda^{-(2-s)/(2+s)}$, bridging smooth and nonsmooth settings. These results have potential implications for nonlinear PDEs with evolving metrics, where metric regularity is limited and orthonormal spectral data are relevant.

Abstract

We establish $L^q$ spectral cluster bounds for families of orthonormal functions associated to the Laplace-Beltrami operator on a compact Riemannian manifold. The metric is only assumed to be of class $C^s$, where $0\leq s\leq 2$.

Theorems & Definitions (2)

• Theorem 1.1: The $C^{1,1}$ case
• Theorem 1.2: The case $s\in[1,2)$] Let $(M,{\rm g})$ be a smooth compact boundaryless Riemannian manifold of dimension $n\geq 2$. Assume that the metric ${\rm g}$ is of class $C^s$ for $s\in[1,2)$ (Lipschitz in case $s=1$). Then there exists $C>0$ such that for any $\lambda\geq 1$, any orthonormal system $(u_j)_{j\in J}\subset E_{\lambda}$ and any sequence $(\nu_j)_{j\in J}\subset {\mathbb C}$, we have, for all $2\leq q\leq\infty$, \|\sum_{j\in J}\nu_j|u_j|^2\|_{L^{q/2}(M)}\leq C\lambda^{2\delta_s(q)}(\sum_{j\in J}|\nu_j|^{\alpha(q)})^{1/\alpha(q)}, where $\alpha(q)$ is given by \ref{['eq-def:alpha']}, and $\delta_s(q):= \frac{n-1}{2}\left(1+\frac{2-s}{2+s}\right)\left(\frac{1}{2}-\frac{1}{q}\right)\text{if}\ 2\leq q\leq q_n,n\left(\frac{1}{2}-\frac{1}{q}\right) -\frac{1}{2}+\frac{2-s}{2+s}\frac{1}{q}\text{if}\ q_n\leq q\leq \infty.$ Moreover, for any cube $Q_R\subset M$ of sidelength $R =\lambda^{-\frac{2-s}{2+s}}$, we have \|\sum_{j\in J}\nu_j|u_j|^2\|_{L^{q/2}(Q_R)}\leq C\lambda^{2\delta(q)}(\sum_{j\in J}|\nu_j|^{\alpha(q)})^{1/\alpha(q)},\quad q_n\leq q\leq \infty, with $\delta(q)$ as in \ref{['eq-def:delta_sogge']}.