, Volume 19, Issue 2, pp 177–220 | Cite as

Newton–Besov spaces and Newton–Triebel–Lizorkin spaces on metric measure spaces

  • Nageswari Shanmugalingam
  • Dachun YangEmail author
  • Wen Yuan


In this paper, via a modification of the notion of weak upper gradients, we introduce and investigate properties of the Newton–Besov spaces \(\textit{NB}^s_{p,q}(X)\) and the Newton–Triebel–Lizorkin spaces \(\textit{NF}^s_{p,q}(X)\), with \(s\in [0,1]\), \(1\le p<\infty \) and \(q\in (0,\infty ]\), of functions on a metric measure space \(X\) and prove that, when \(1<p<\infty \), the space \(\textit{NB}^1_{p,\infty }(X)\) coincides with the Newton–Sobolev space \(N^{1,p}(X)\). A Poincaré type inequality related to these function spaces is also investigated. Sensitivity to changes of functions in these classes on sets of measure zero is also demonstrated. Even in the Euclidean setting \(X={\mathbb R}^n\), these results are also new.


Newton–Besov space Newton–Triebel–Lizorkin space  Newton–Sobolev space Upper gradient Hajłasz gradient Modulus  Poincaré inequality Semmes pencil of curves 

Mathematics Subject Classification (2010)

Primary 42B35 Secondary 46E35 30L99 



The authors would like to thank the referee for her/his very careful reading and helpful comments which improve the presentation of this article.


  1. 1.
    Aikawa, H., Ohtsuka, M.: Extremal length of vector measures. Ann. Acad. Sci. Fenn. Math. 24, 61–88 (1999)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bahouri, H., Chemin, J.-Y.: Équations d’ondes quasilinéaires et estimations de Strichartz. Am. J. Math. 121, 1337–1377 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bennett, C., Sharpley, R.: Interpolation of Operators, Pure and Applied Mathematics 129, pp. xiv+469. Academic Press, Boston (1988)Google Scholar
  4. 4.
    Besov, O.V.: On some families of function spaces. Embedding and extension theorems. Dokl. Akad. Nauk SSSR 126, 1163–1165 (1959)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Besov, O.V.: Investigation of a class of function spaces in connection with imbedding and extension theorems. Trudy. Mat. Inst. Steklov. 60, 42–81 (1961)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Björn, A., Björn, J., Gill, J., Shanmugalingam, N.: Geometric analysis on Cantor sets and trees. arXiv:1304.0566
  7. 7.
    Bojarski, B.: Remarks on Some Geometric Properties of Sobolev Mappings. In: Functional Analysis and Related Topics (Sapporo, 1990), pp. 65–76. World Sci. Publ., River Edge (1991)Google Scholar
  8. 8.
    Bojarski, B., Hajłasz, P.: Pointwise inequalities for Sobolev functions and some applications. Stud. Math. 106, 77–92 (1993)zbMATHGoogle Scholar
  9. 9.
    Bourdon, M., Pajot, H.: Cohomologie \(l_p\) et espaces de Besov. J. Reine Angew. Math. 558, 85–108 (2003)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Bourgain, J., Pavlović, N.: Ill-posedness of the Navier–Stokes equations in a critical space in 3D. J. Funct. Anal. 255, 2233–2247 (2008)Google Scholar
  11. 11.
    Buckley, S.: Is the maximal function of a Lipschitz function continuous? Ann. Acad. Sci. Fenn. Math. 24, 519–528 (1999)MathSciNetGoogle Scholar
  12. 12.
    Caffarelli, L., Roquejoffre, J.-M., Sire, Y.: Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. (JEMS) 12, 1151–1179 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Caffarelli, L., Roquejoffre, J.-M., Savin, O.: Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63, 1111–1144 (2010)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Chae, D.: On the well-posedness of the Euler equations in the Triebel–Lizorkin spaces. Commun. Pure Appl. Math. 55, 654–678 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Chae, D.: On the Euler equations in the critical Triebel–Lizorkin spaces. Arch. Ration. Mech. Anal. 170, 185–210 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Danchin, R.: Global existence in critical spaces for compressible Navier–Stokes equations. Invent. Math. 141, 579–614 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Franchi, B., Hajłasz, P., Koskela, P.: Definitions of Sobolev classes on metric spaces. Ann. Inst. Fourier (Grenoble) 49, 1903–1924 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Fuglede, B.: Extremal length and functional completion. Acta. Math. 98, 171–218 (1957)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Gogatishvili, A., Koskela, P., Shanmugalingam, N.: Interpolation properties of Besov spaces defined on metric spaces. Math. Nachr. 283, 215–231 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Gogatishvili, A., Koskela, P., Zhou, Y.: Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces. Forum Math. 25, 787–819 (2013)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Grafakos, L., Liu, L., Yang, D.: Vector-valued singular integrals and maximal functions on spaces of homogeneous type. Math. Scand. 104, 296–310 (2009)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Hajłasz, P.: Sobolev spaces on an arbitrary metric spaces. Potential Anal. 5, 403–415 (1996)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Hajłasz, P.: Sobolev Spaces on Metric-Measure Spaces. In: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), pp. 173–218. Contemp. Math. 338, Am. Math. Soc., Providence (2003)Google Scholar
  24. 24.
    Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688), x+101 (2000)Google Scholar
  25. 25.
    Han, Y., Müller, D., Yang, D.: A theory of Besov and Triebel–Lizorkin spaces on metric measure spaces modeled on Carnot–Carathéodory spaces. Abstr. Appl. Anal. 893409, 250 (2008)Google Scholar
  26. 26.
    Haroske, D.D., Triebel, H.: Embeddings of function spaces: a criterion in terms of differences. Complex Var. Elliptic Equ. 56, 931–944 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)zbMATHCrossRefGoogle Scholar
  28. 28.
    Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.T.: Sobolev classes of Banach space-valued functions and quasiconformalmappings. J. Anal. Math. 85, 87–139 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Hu, J.: A note on Hajłasz–Sobolev spaces on fractals. J. Math. Anal. Appl. 280, 91–101 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Jonsson, A., Wallin, H.: Boundary value problems and Brownian motion on fractals. Chaos Solitons Fractals 8, 191–205 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Jonsson, A., Wallin, H.: Function spaces on subsets of \(\mathbb{R}^{n}\). Math. Rep. 2(1), xiv+221 (1984)Google Scholar
  33. 33.
    Koskela, P., MacManus, P.: Quasiconformal mappings and Sobolev spaces. Studia Math. 131, 1–17 (1998)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Koskela, P., Saksman, E.: Pointwise characterizations of Hardy–Sobolev functions. Math. Res. Lett. 15, 727–744 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Koskela, P., Yang, D., Zhou, Y.: A characterization of Hajłasz–Sobolev and Triebel–Lizorkin sapces via grand Littlewood–Paley functions. J. Funct. Anal. 258, 2637–2661 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Koskela, P., Yang, D., Zhou, Y.: Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings. Adv. Math. 226, 3579–3621 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    McShane, E.J.: Extension of range of functions. Bull. Am. Math. Soc. 40, 837–842 (1934)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Müller, D., Yang, D.: A difference characterization of Besov and Triebel–Lizorkin spaces on RD-spaces. Forum Math. 21, 259–298 (2009)Google Scholar
  39. 39.
    Ohtsuka, M.: Extremal Length and Precise Functions, GAKUTO International Series, Mathematical Sciences and Applications 19, pp. vi+343. Gakkōtosho Co., Ltd., Tokyo (2003)Google Scholar
  40. 40.
    Semmes, S.: Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities. Selecta Math. (N. S.) 2, 155–295 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16, 243–279 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Shanmugalingam, N.: Harmonic functions on metric spaces. Illinois J. Math. 45, 1021–1050 (2001)zbMATHMathSciNetGoogle Scholar
  43. 43.
    Takada, R.: Counterexamples of commutator estimates in the Besov and the Triebel–Lizorkin spaces related to the Euler equations. SIAM J. Math. Anal. 42, 2473–2483 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)CrossRefGoogle Scholar
  45. 45.
    Triebel, H.: Theory of Function Spaces II. Birkhäuser, Basel (1992)zbMATHCrossRefGoogle Scholar
  46. 46.
    Triebel, H.: Fractals and Spectra, Related to Fourier Analysis and Function Spaces, pp. viii+271. Birkhäuser, Basel (1997)Google Scholar
  47. 47.
    Triebel, H.: Theory of Function Spaces III, pp. xii+426. Birkhäuser, Basel (2006)Google Scholar
  48. 48.
    Triebel, H.: Sobolev–Besov spaces of measurable functions. Stud. Math. 201, 69–86 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Triebel, H.: Limits of Besov norms. Arch. Math. 96, 169–175 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Väisäla, J.: Lectures on \(n\)-dimensional Quasiconformal Mapping, Lecture notes in Mathematics 229, pp. xiv+144. Springer, Berlin, New York (1971)Google Scholar
  51. 51.
    Vishik, M.: Hydrodynamics in Besov spaces. Arch. Rational Mech. Anal. 145, 197–214 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Yang, D.: New characterizations of Hajłasz–Sobolev spaces on metric spaces. Sci. China Ser. A 46, 675–689 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Yang, D., Zhou, Y.: New properties of Besov and Triebel–Lizorkin spaces on RD-spaces. Manuscripta Math. 134, 59–90 (2011)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Nageswari Shanmugalingam
    • 1
  • Dachun Yang
    • 2
    Email author
  • Wen Yuan
    • 2
  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA
  2. 2.School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex SystemsMinistry of EducationBeijingPeople’s Republic of China

Personalised recommendations