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 R C D \mathsf {RCD} spaces. Though that condition is seemed to always hold for any such spaces, however, Dai, Honda, Pan, and Wei recently showed that for any integer n n at least 2, there exists a compact R C D \mathsf {RCD} space of n n dimension fails to satisfy the regular Weyl’s law. In this short article we prove that one dimensional R C D \mathsf {RCD} spaces always satisfy the regular Weyl’s law.