Sobolev and BV functions on RCD spaces via the short-time behavior of the heat kernel

In the setting of finite-dimensional [Formula: see text] spaces, we characterize the [Formula: see text]-Sobolev spaces for [Formula: see text] and the space of functions of bounded variation in terms of the short-time behavior of the heat flow. Moreover, we prove that Cheeger [Formula: see text]-energies and total variations can be computed as limits of nonlocal functionals involving the heat kernel.