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In this paper, we prove pointwise convergence of heat kernels formGH-convergent sequences of $RCD^*(K,N)$-spaces. We obtain as a corollaryresults on the short-time behavior of the heat kernel in $RCD^*(K,N)$-spaces.We use then these results to initiate the study of Weyl's law in the $RCD$setting

Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces