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In the setting of finite-dimensional $\mathrm{RCD}(K,N)$ spaces, wecharacterize the $p$-Sobolev spaces for $p\in(1,\infty)$ and the space offunctions of bounded variation in terms of the short-time behaviour of the heatflow. Moreover, we prove that Cheeger $p$-energies and total variations can becomputed as limits of nonlocal functionals involving the heat kernel.

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