A new characterization of Sobolev spaces on Lipschitz differentiability spaces

Numerous characterizations of Sobolev norms via the asymptotic behavior of non-local functionals have been established over the past decades; however, their validity beyond the PI framework remains poorly understood. We establish such a characterization on Lipschitz differentiability spaces without assuming either the doubling condition or a Poincar\'e inequality, by proving sharp two-sided Brezis--Van Schaftingen--Yung type asymptotic formulas. We also construct sharp counterexamples revealing the necessity of our assumptions, and provide several examples which are of independent interest.