On local trees with uniform volume growth, the intrinsic Sobolev and BV norms are equivalent to Korevaar-Schoen energies and to heat-kernel Besov norms.
Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates
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Korevaar-Schoen and heat kernel characterizations of Sobolev and BV spaces on local trees
On local trees with uniform volume growth, the intrinsic Sobolev and BV norms are equivalent to Korevaar-Schoen energies and to heat-kernel Besov norms.