A removability theorem for Sobolev functions and detour sets

  • Dimitrios NtalampekosEmail author


We study the removability of compact sets for continuous Sobolev functions. In particular, we focus on sets with infinitely many complementary components, called “detour sets”, which resemble the Sierpiński gasket. The main theorem is that if \(K \subset \mathbb {R}^n\) is a detour set and its complementary components are sufficiently regular, then K is \(W^{1,p}\)-removable for \(p>n\). Several examples and constructions of sets where the theorem applies are given, including the Sierpiński gasket, Apollonian gaskets, and Julia sets.


Removability Sobolev functions Hölder domains Detour sets Sierpiński gasket 

Mathematics Subject Classification

Primary 46E35 Secondary 30C65 



The author would like to thank Mario Bonk for introducing him to the problem of removability, and for several fruitful discussions and explanations on the background of the problem and the proofs of previous results. Additional thanks go to Huy Tran for pointing out the connection of the problem to SLE, to Ville Tengvall for pointing out the reference [14], and to Pekka Koskela for a motivating discussion. The author is also grateful to Vasiliki Evdoridou, Malik Younsi, and the anonymous referee for their comments and corrections. This paper was written while the author was visiting University of Helsinki. He thanks the faculty and staff of the Department of Mathematics at the University of Helsinki for their hospitality.


  1. 1.
    Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
  2. 2.
    Benedetto, J., Czaja, W.: Integration and modern analysis. Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser, Boston, Inc., Boston, MA (2009)Google Scholar
  3. 3.
    Bishop, C.: Some homeomorphisms of the sphere conformal off a curve. Ann. Acad. Sci. Fenn. Ser. A I Math. 19(2), 323–338 (1994)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI (2001)Google Scholar
  5. 5.
    Devaney, R., Rocha, M., Siegmund, S.: Rational maps with generalized Sierpiński gasket Julia sets. Topol. Appl. 154(1), 11–27 (2007)CrossRefGoogle Scholar
  6. 6.
    Gehring, F.W.: The \(L^p\)-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer Verlag, New York (2001)CrossRefGoogle Scholar
  8. 8.
    Jones, P.: On removable sets for Sobolev spaces in the plane. Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), 250–276, Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ (1995)Google Scholar
  9. 9.
    Jones, P., Smirnov, S.: Removability theorems for Sobolev functions and quasiconformal maps. Ark. Mat. 38(2), 263–279 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kameyama, A.: Julia sets of postcritically finite rational maps and topological self-similar sets. Nonlinearity 13(1), 165–188 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kaufman, R.: Fourier-Stieltjes coefficients and continuation of functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 9, 27–31 (1984)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Koskela, P.: Removable sets for Sobolev spaces. Ark. Mat. 37(2), 291–304 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Koskela, P., Nieminen, T.: Quasiconformal removability and the quasihyperbolic metric. Indiana Univ. Math. J. 54(1), 143–151 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Koskela, P., Rajala, T., Zhang, Y.: A density problem for Sobolev functions on Gromov hyperbolic domains. Nonlinear Anal. 154, 189–209 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Martio, O., Väisälä, J.: Quasihyperbolic geodesics in convex domains II. Pure Appl. Math. Q. 7(2), Special Issue: In honor of Frederick W. Gehring, Part 2, 395–409 (2011)Google Scholar
  16. 16.
    Maz’ya, V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, vol. 324, Springer-Verlag, Berlin Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Mihalache, N.: Julia and John revisited. Fund. Math. 215(1), 67–86 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Milnor, J.: Dynamics in one complex variable. Third edition, Annals of Mathematical Studies, vol. 160, Princeton University Press, Princeton (2006)Google Scholar
  19. 19.
    Newman, M.H.A.: Elements of the topology of plane sets of points, 2nd edn. Cambridge University Press, London (1964)Google Scholar
  20. 20.
    Ntalampekos, D., Wu, J.-M.: Non-removability of Sierpiński spaces. Proc. Am. Math. Soc. (2019), to appear,
  21. 21.
    Ntalampekos, D.: Non-removability of Sierpiński carpets, Indiana Univ. Math. J. (2019), to appear. Preprint arXiv:1809.05605
  22. 22.
    Ntalampekos, D.: Non-removability of the Sierpiński gasket. Invent. Math. 216(2), 519–595 (2019)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Sheffield, S.: Conformal weldings on random surfaces: SLE and the quantum gravity zipper. Ann. Probab. 44(5), 3474–3545 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Smith, W., Stegenga, D.: Hölder domains and Poincaré domains. Trans. Am. Math. Soc. 319(1), 67–100 (1990)zbMATHGoogle Scholar
  25. 25.
    Väisälä, J.: Lectures on \(n\)-dimensional quasiconformal mappings. In: Lecture Notes in Mathematics, vol. 229, Springer-Verlag, Berlin-New York (1971)Google Scholar
  26. 26.
    Whyburn, G.T.: Analytic Topology. In: American Mathematical Society Colloquium Publications, vol. 28, American Mathematical Society, New York (1942)Google Scholar
  27. 27.
    Younsi, M.: On removable sets for holomorphic functions. EMS Surv. Math. Sci. 2(2), 219–254 (2015)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Institute for Mathematical SciencesStony Brook UniversityStony BrookUSA

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