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

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.