Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Reconstructions of distances by energy forms

Author: Shin-ichi Ohta
Journal: Proc. Amer. Math. Soc. 134 (2006), 3405-3415
MSC (2000): Primary 58C05, 53C60
Published electronically: May 8, 2006
MathSciNet review: 2231926
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that, if a metric measure space admits a stratification so that each stratum satisfies the strong doubling condition, then the intrinsic distance induced from the Cheeger-type energy form coincides with the original distance. In other words, we can reconstruct the distance function by the Cheeger-type energy form. We also observe that this reconstruction does not work for the Korevaar-Schoen-type energy form.

References [Enhancements On Off] (What's this?)

  • [BM] M. Biroli and U. Mosco, A Saint-Venant type principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl. 169 (1995), 125-181. MR 1378473 (97b:35082)
  • [C] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428-517. MR 1708448 (2000g:53043)
  • [FOT] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics 19, Walter de Gruyter & Co., Berlin, 1994. MR 1303354 (96f:60126)
  • [HK] P. Haj\lasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688. MR 1683160 (2000j:46063)
  • [He] J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917 (2002c:30028)
  • [HeK] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1-61.MR 1654771 (99j:30025)
  • [KoSc] N. J. Korevaar and R. M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), 561-659.MR 1266480 (95b:58043)
  • [KMS] K. Kuwae, Y. Machigashira, and T. Shioya, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), 269-316.MR 1865418 (2002m:58052)
  • [KuSh] K. Kuwae and T. Shioya, On generalized measure contraction property and energy functionals over Lipschitz maps, Potential Anal. 15 (2001), 105-121.MR 1838897 (2002f:31022)
  • [O] S. Ohta, Regularity of harmonic functions in Cheeger-type Sobolev spaces, Ann. Global Anal. Geom. 26 (2004), 397-410.MR 2103408 (2005h:35053)
  • [R1] A. Ranjbar-Motlagh, A note on the Poincaré inequality, Studia Math. 154 (2003), 1-11. MR 1949045 (2004b:46039)
  • [R2] A. Ranjbar-Motlagh, Poincaré inequality for abstract spaces, Bull. Austral. Math. Soc. 71 (2005), 193-204. MR 2133404
  • [S] K.-T. Sturm, Diffusion processes and heat kernels on metric spaces, Ann. Probab. 26 (1998), 1-55. MR 1617040 (99b:31008)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58C05, 53C60

Retrieve articles in all journals with MSC (2000): 58C05, 53C60

Additional Information

Shin-ichi Ohta
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan

Keywords: Intrinsic distance, Cheeger-type energy form, Korevaar-Schoen-type energy form, strong doubling condition, geodesic bicombing
Received by editor(s): November 17, 2004
Received by editor(s) in revised form: May 25, 2005
Published electronically: May 8, 2006
Additional Notes: This work was partially supported by the Grant-in-Aid for Scientific Research for Young Scientists (B) 16740034 from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society