Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm

Journal of Functional Analysis

Volumn 263, Issue 4, 2012, Pages 896-924

  Rajala, Tapio ^a,b  

^a SCUOLA NORMALE SUPERIORE   (Italy)

^b UNIVERSITY OF JYVÄSKYLÄ   (Finland)

Author keywords

Measure contraction property; Metric measure space; Poincar inequality; Ricci curvature

Indexed keywords

EID: 84861980324     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2012.05.006     Document Type: Article

References (24)

1. Ambrosio L., Gigli N., Savaré G. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures Math. ETH Zürich 2008, Birkhäuser Verlag, Basel. second ed.
2. L. Ambrosio, N. Gigli, G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, preprint, 2011.
3. Bacher K., Sturm K.-T. Localization and tensorization properties of the curvature-dimension condition for metric measure spaces. J. Funct. Anal. 2010, 259:28-56.
4. Buser P. A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. 1982, 15:213-230.
5. Cheeger J. Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 1999, 9:428-517.
6. Cheeger J., Colding T. On the structure of spaces with Ricci curvature bounded below. III. J. Differential Geom. 2000, 54:37-74.
7. N. Gigli, Optimal maps in non branching spaces with Ricci curvature bounded from below, Geom. Funct. Anal., in press.
8. Hajłasz P., Koskela P. Sobolev meets Poincaré. C. R. Acad. Sci. Paris Sér. I Math. 1995, 320:1211-1215.
9. Hajłasz P., Koskela P. Sobolev met Poincaré. Mem. Amer. Math. Soc. 2000, 145(688).
10. Heinonen J. Lectures on Analysis on Metric Spaces 2001, Springer-Verlag, New York.
11. Heinonen J., Koskela P. Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 1998, 181:1-61.
12. Kantorovich L.V. On the translocation of masses. C. R. (Dokl.) Acad. Sci. URSS 1942, 37:199-201.
13. Kantorovich L.V. On a problem of Monge. Uspekhi Mat. Nauk 1948, 3:225-226. (in Russian).
14. Lott J., Villani C. Weak curvature conditions and functional inequalities. J. Funct. Anal. 2007, 245:311-333.
15. Lott J., Villani C. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 2009, 169:903-991.
16. Monge G. Mémoire sur la théorie des déblais et remblais. Mémoire de l'académie des sciences de Paris 1781.
17. Ohta S.-i. On the measure contraction property of metric measure spaces. Comment. Math. Helv. 2007, 82:805-828.
18. Rajala T. Local Poincaré inequalities from stable curvature conditions on metric spaces. Calc. Var. Partial Differential Equations 2012, 44:477-494.
19. T. Rajala, Improved geodesics for the reduced curvature-dimension condition in branching metric spaces, Discrete Contin. Dyn. Syst., in press; . arxiv:1202.6514.
20. von Renesse M.-K. On local Poincaré via transportation. Math. Z. 2008, 259:21-31.
21. Semmes S. Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities. Selecta Math. 1996, 2:155-295.
22. Sturm K.-T. On the geometry of metric measure spaces. I. Acta Math. 2006, 196:65-131.
23. Sturm K.-T. On the geometry of metric measure spaces. II. Acta Math. 2006, 196:133-177.
24. Villani C. Optimal Transport. Old and New. Grundlehren Math. Wiss. 2009, vol. 338. Springer-Verlag, Berlin.