We prove that on an arbitrary metric measure space a countable collection of test plans is sufficient to recover all BV functions and their total variation measures. In the setting of non-branching CD(K, N ) spaces (with finite reference measure), we can additionally require these test plans to be concentrated on geodesics.

We show that in a bounded Gromov hyperbolic domain Ω smooth functions with bounded derivatives C ∞ (Ω) ∩ W k,∞ (Ω) are dense in the homogeneous Sobolev spaces L k,p (Ω). IntroductionWe continue the study of density of functions with bounded derivatives in the space of Sobolev functions in a domain in R n . It was shown by Koskela-Zhang [13] that for a simply connected planar domain Ω ⊂ R 2 , W 1,∞ is dense in W 1,p and in the special case of Jordan domains also C ∞ (R 2 ) ∩ W 1,∞ (Ω) is dense. The above result of Koskela-Zhang has been generalized to have the density of W k,∞ in the homogeneous Sobolev space L k,p for k ∈ N in planar simply connected domains by Nandi-Rajala-Schultz [20]. In dimensions higher than two however simply connectedness is not sufficient (see for example [14]). Recall that simply connected planar domains are negatively curved in the (quasi) hyperbolic metric. A useful metric generalization of negatively curved spaces was introduced by Gromov [8], in the context of group theory. Following Bonk-Heinonen-Koskela [4], we call a domain Gromov hyperbolic if, when equipped with the quasihyperbolic metric, it is δ-hyperbolic in the sense of Gromov, for some δ ≥ 0 (see Section 3 for definitions). Gromov hyperbolicity has turned out to be a sufficient condition for the density of W 1,∞ in W 1,p , as shown by Koskela-Rajala-Zhang [14], and these are primarily the domains we consider in this paper.Let us mention here a few similarities between our setting and the planar simply connected case. We recall that simply connected domains with the quasihyperbolic metric (equivalent to the hyperbolic metric by the Koebe distortion theorem) in the plane are Gromov hyperbolic and that the quasihyperbolic geodesics are unique (see Luiro [17]); conversely, a Gromov hyperbolic domain with uniqueness of quasihyperbolic geodesics is simply connected. The latter is of course true in higher dimensions as well (indeed, in this case, for any pair of points, any curve γ joining the points is homotopic to the unique quasihyperbolic geodesic Γ joining the given points, with the homotopy given by quasihyperbolic geodesics joining the points γ(t) and Γ(t) once Γ is parametrized suitably). Gromov hyperbolic domains are often seen as a topological generalization of planar simply connected domains. It was shown by Bonk-Heinonen-Koskela [4] that Gromov hyperbolic domains are conformally equivalent to suitable uniform metric spaces equipped with the quasihyperbolic metric and corresponding suitable measures. Finally, we note that in higher dimensions, simply connectedness alone does not imply Gromov hyperbolicity; consider for example the unit ball in R 3 deformed to have a cuspidal-wedge along an equator.

In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corollary we obtain that if a metric measure space is a very strict C D ( K , N ) CD(K,N) -space or an essentially non-branching M C P ( K , N ) MCP(K,N) -space with some open set isometric to an interval, then it is a one-dimensional manifold. We also obtain the same conclusion for a metric measure space which has a point in which the Gromov-Hausdorff tangent is unique and isometric to the real line, and for which the optimal transport maps not only exist but are unique. Again, we obtain an analogous corollary in the setting of essentially non-branching M C P ( K , N ) MCP(K,N) -spaces.

We introduce a more restrictive version of the strict CD(K, ∞) -condition, the so-called very strict CD(K, ∞) -condition, and show the existence of optimal maps in very strict CD(K, ∞) -spaces despite the possible lack of uniqueness of optimal plans.

The relation between the subclass x °f balanced one-sided ideals, which contains all two-sided ideals, and weakly prime one-sided ideals is considered. If a member P of x is prime, then it is prime in its idealizer I(P) . Furthermore, if the ring is left and right noetherian, then I(P) modulo P is a prime Goldie ring.