One dimensional RCD spaces always satisfy the regular Weyl's law

TL;DR

This paper proves that all one-dimensional RCD spaces satisfy the regular Weyl's law, filling a gap in understanding the spectral properties of these spaces across dimensions.

Contribution

It establishes that one-dimensional RCD spaces always meet the conditions for the regular Weyl's law, contrasting with higher-dimensional cases.

Findings

One-dimensional RCD spaces satisfy the regular Weyl's law.

Higher-dimensional RCD spaces can fail to satisfy the law.

Clarifies the dimensional dependence of spectral properties in RCD spaces.

Abstract

Ambrosio, Honda, and Tewodrose proved that the regular Weyl's law is equivalent to a mild condition related to the infinitesimal behavior of the measure of balls in compact finite dimensional RCD spaces. Though that condition is seemed to always hold for any such spaces, however, Dai, Honda, Pan, and Wei recently show that for any integer n at least 2, there exists a compact RCD space of n dimension fails to satisfy the regular Weyl's law. In this short article we prove that one dimensional RCD spaces always satisfy the regular Weyl's law.