Equality of D1,p(X) and N1,p(X)+R for spaces with a single p-hyperbolic end

Determine whether the Dirichlet space D1,p(X) coincides with the Newton–Sobolev space N1,p(X) + R for metric measure spaces (X, d, μ) satisfying the paper’s standing assumptions (complete, connected, proper, separable, μ(X)=∞, and 0 < μ(B(x,r)) < ∞ for all x and r) and uniformly locally p-controlled geometry, in the case where X has exactly one end at infinity and that end is p-hyperbolic. Clarify whether equality holds, fails, or depends on additional geometric or analytic conditions in this single p-hyperbolic end setting.

Background

The paper studies when the homogeneous Newton–Sobolev space N1,p(X)+R agrees with the Dirichlet space D1,p(X) on metric measure spaces with uniformly locally p-controlled geometry. It establishes many positive and negative results depending on global doubling, Gromov hyperbolicity and uniformization, parabolicity, and the structure of ends at infinity.

Theorem 1.1 resolves large classes: equality holds for spaces of globally p-controlled geometry, certain Gromov hyperbolic uniformizations when p exceeds a threshold, and spaces with parabolic behavior or multiple ends yield non-equality. Section 6 further treats spaces with multiple ends and parabolic ends, and Section 8 fully resolves the standard hyperbolic space H^n, proving equality precisely for 1 ≤ p ≤ n−1.

What remains unresolved in general is the case of spaces with exactly one end at infinity that is p-hyperbolic. The paper pinpoints this as the only unknown situation left after the preceding results and then analyzes H^n as an example, but a general characterization for all such spaces is not established.