Short-time behavior of the heat kernel and Weyl's law on \(\mathrm{RCD}^*(K,N)\) spaces (Q1691983)

This paper presents a wide generalization of Weyl law for the asymptotics of eigenvalues of the Laplacian to the context of the class of metric measure spaces known as \(\mathrm{RCD}^*(K,N)\) spaces, where the growth rate of the eigenvalues, \(\lambda^{\frac{k}{2}}\), is determined as the the maximal integer \(k\) for which the \(k\)-dimensional regular set has a positive measure. The Weyl law is obtained as a consequence of results on the short-time behavior of the heat kernel, which are in turn obtained from a result about pointwise convergence of heat kernels of \(\mathrm{RCD}^*(K,N)\) under measured Gromov-Hausdorff convergence of such spaces.