Density of bounded Lipschitz functions in the nonlocal Sobolev space X2 on general metric spaces

Determine whether the density property Assumption (Ass:F-bis) holds beyond Euclidean settings: specifically, ascertain if bounded Lipschitz functions are dense in the nonlocal Sobolev space X2(V,d,π,κ) = {φ ∈ L∞(V;π) : ∫∫_{V×V} |φ(y)−φ(x)|² (π(dx) κ(x,dy)) < ∞} on general separable metric measure spaces (V,d,π) equipped with singular jump kernels κ satisfying the measurability and integrability conditions sup_{x∈V} ∫_V (1∧d²(x,y)) κ(x,dy) < ∞ and detailed balance. Concretely, show whether for every φ ∈ X2 there exists a sequence (φ_n)_n ⊂ Lip(V) with φ_n → φ in L¹(V;π) and (φ_n(y)−φ_n(x)) → (φ(y)−φ(x)) in L²(V×V; π(dx)κ(x,dy)).

Background

The paper introduces reflecting solutions for generalized gradient-flow formulations of singular jump processes. A key tool is a reflecting continuity equation tested against functions in the space X2, defined by square-integrability of the discrete gradient with respect to the coupling measure π(dx)κ(x,dy).

Upgrading solutions that satisfy the weak continuity equation with Lipschitz test functions to reflecting solutions requires the density of bounded Lipschitz functions in X2 (Assumption (Ass:F-bis)). This density can be proved in classical settings such as R^d, bounded domains, and the torus via convolution methods, but may fail in certain singular configurations, as illustrated by examples in the appendix.

The authors note that their existence and robustness results do not rely on this density property; however, uniqueness and the chain rule for the reflecting formulation benefit from it. The broader validity of this density property in general metric spaces with singular kernels remains unresolved.