TY - JOUR AB - We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature-dimension condition recently introduced in Milman (Commun Pure Appl Math, to appear). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat semigroup. These results apply in particular to large classes of (ideal) sub-Riemannian manifolds. AU - Dello Schiavo, Lorenzo AU - Suzuki, Kohei ID - 10588 JF - Mathematische Annalen KW - quasi curvature-dimension condition KW - sub-riemannian geometry KW - Sobolev-to-Lipschitz property KW - Varadhan short-time asymptotics SN - 0025-5831 TI - Sobolev-to-Lipschitz property on QCD- spaces and applications VL - 384 ER -