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.

This study includes a series of 5 cases demonstrating cardiovascular manifestations of findings of IgG4-related disease. Two cases demonstrate peri-aortic soft tissue thickening in the infrarenal abdominal aorta and bilateral ostio--proximal common iliac artery. In 2 cases there were circumferential soft tissue lesions around the arch of the aorta. One of the cases showed coexistent, biopsy-proven Riedel’s thyroiditis and infiltrative soft tissue along the right atrial wall and interatrial septum. In one case there was a partly calcified mass in the left hemi-thorax consistent with a diagnosis of IgG4-related fibrosing mediastinitis.

For a bounded simply connected domain Ω ⊂ R 2 , any point z ∈ Ω and any 0 < α < 1, we give a lower bound for the αdimensional Hausdorff content of the set of points in the boundary of Ω which can be joined to z by a John curve with a suitable John constant depending only on α, in terms of the distance of z to ∂Ω. In fact this set in the boundary contains the intersection ∂Ω z ∩ ∂Ω of the boundary of a John sub-domain Ω z of Ω, centered at z, with the boundary of Ω. This may be understood as a quantitative version of a result of Makarov. This estimate is then applied to obtain the pointwise version of a weighted Hardy inequality.

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