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)



Menger curvature and $C^{1}$ regularity of fractals

Authors: Yong Lin and Pertti Mattila
Journal: Proc. Amer. Math. Soc. 129 (2001), 1755-1762
MSC (2000): Primary 28A75
Published electronically: October 31, 2000
MathSciNet review: 1814107
Full-text PDF

Abstract | References | Similar Articles | Additional Information


We show that if $E$ is an $s$-regular set in $\mathbf{R}^{2}$ for which the triple integral $\int _{E}\int _{E}\int _{E}c(x,y,z)^{2s}\,d\mathcal{H}^{s}x\,d\mathcal{H}^{s}y\,d \mathcal{H}^{s}z$of the Menger curvature $c$ is finite and if $0<s\le 1/2$, then $\mathcal{H}^{s}$almost all of $E$ can be covered with countably many $C^{1}$ curves. We give an example to show that this is false for $1/2<s<1$.

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

  • [D] G. David, Unrectifiable $1$-sets have vanishing analytic capacity, Rev. Mat. Iberoamericana 14 (1998), 369-479. MR 99i:42018
  • [DS] G. David and S. Semmes, Singular integrals and rectifiable sets in $\mathbf{R}^n$: au-delà des graphes lipschitziens, Astérisque 193 (1991). MR 92j:42016
  • [F] H. Federer, Geometric Measure Theory, Springer-Verlag, 1969. MR 41:1976
  • [K] S. Kass, Karl Menger, Notices Amer. Math. Soc. 43:5 (1996), 558-561. CMP 96:12
  • [L] J.-C. Léger, Menger curvature and rectifiability, Ann. of Math. 149 (1999), 831-869. CMP 99:17
  • [Li] Y. Lin, Menger curvature, singular integrals and analytic capacity, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 111, 1997. MR 98j:30020
  • [MM] M. A. Martin and P. Mattila, $k$-dimensional regularity classifications for $s$-fractals, Trans. Amer. Math. Soc. 305 (1988), 293-315. MR 88j:28007
  • [M1] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995. MR 96h:28006
  • [M2] P. Mattila, Rectifiability, analytic capacity, and singular integrals, Doc. Math. Extra Volume ICM 1998, II, 657-664. MR 99g:28010
  • [MMV] P. Mattila, M. S. Melnikov and J. Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math. 144 (1996), 127-136. MR 97k:31004
  • [Me] M. S. Melnikov, Analytic capacity: discrete approach and curvature of measure, Sb. Math. 186 (1995), 57-76. MR 96f:30020
  • [MV] M. Melnikov and J. Verdera, A geometric proof of the $L^{2}$ boundedness of the Cauchy integral on Lipschitz graphs, Internat. Math. Res. Notices 7 (1995), 325-331. MR 96f:45011

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 28A75

Retrieve articles in all journals with MSC (2000): 28A75

Additional Information

Yong Lin
Affiliation: Department of Mathematics, Renmin University of China, Information School, Beijing, 100872, China

Pertti Mattila
Affiliation: Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland

Received by editor(s): September 27, 1999
Published electronically: October 31, 2000
Additional Notes: The authors gratefully acknowledge the hospitality of CRM at Universitat Autònoma de Barcelona where part of this work was done. The first author also wants to thank the Academy of Finland for financial support.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society