Sobolev and BV functions on $\mathrm{R}\mathrm{C}\mathrm{D}$ spaces via the short-time behaviour of the heat kernel

In the setting of finite-dimensional $\mathrm{R}\mathrm{C}\mathrm{D}(K,N)$ spaces, we characterize the $p$-Sobolev spaces for $p\in (1,\mathrm{\infty })$ and the space of functions of bounded variation in terms of the short-time behaviour of the heat flow. Moreover, we prove that Cheeger $p$-energies and total variations can be computed as limits of nonlocal functionals involving the heat kernel.