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

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Abstract

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.

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References

  1. Aikawa, H., Ohtsuka, M.: Extremal length of vector measures. Ann. Acad. Sci. Fenn. Math. 24, 61–88 (1999)

    MATH  MathSciNet  Google Scholar 

  2. Bahouri, H., Chemin, J.-Y.: Équations d’ondes quasilinéaires et estimations de Strichartz. Am. J. Math. 121, 1337–1377 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bennett, C., Sharpley, R.: Interpolation of Operators, Pure and Applied Mathematics 129, pp. xiv+469. Academic Press, Boston (1988)

  4. Besov, O.V.: On some families of function spaces. Embedding and extension theorems. Dokl. Akad. Nauk SSSR 126, 1163–1165 (1959)

    MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  6. Björn, A., Björn, J., Gill, J., Shanmugalingam, N.: Geometric analysis on Cantor sets and trees. arXiv:1304.0566

  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)

  8. Bojarski, B., Hajłasz, P.: Pointwise inequalities for Sobolev functions and some applications. Stud. Math. 106, 77–92 (1993)

    MATH  Google Scholar 

  9. Bourdon, M., Pajot, H.: Cohomologie \(l_p\) et espaces de Besov. J. Reine Angew. Math. 558, 85–108 (2003)

    MATH  MathSciNet  Google Scholar 

  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)

  11. Buckley, S.: Is the maximal function of a Lipschitz function continuous? Ann. Acad. Sci. Fenn. Math. 24, 519–528 (1999)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  13. Caffarelli, L., Roquejoffre, J.-M., Savin, O.: Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63, 1111–1144 (2010)

    MATH  MathSciNet  Google Scholar 

  14. Chae, D.: On the well-posedness of the Euler equations in the Triebel–Lizorkin spaces. Commun. Pure Appl. Math. 55, 654–678 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chae, D.: On the Euler equations in the critical Triebel–Lizorkin spaces. Arch. Ration. Mech. Anal. 170, 185–210 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Danchin, R.: Global existence in critical spaces for compressible Navier–Stokes equations. Invent. Math. 141, 579–614 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Franchi, B., Hajłasz, P., Koskela, P.: Definitions of Sobolev classes on metric spaces. Ann. Inst. Fourier (Grenoble) 49, 1903–1924 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Fuglede, B.: Extremal length and functional completion. Acta. Math. 98, 171–218 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gogatishvili, A., Koskela, P., Shanmugalingam, N.: Interpolation properties of Besov spaces defined on metric spaces. Math. Nachr. 283, 215–231 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gogatishvili, A., Koskela, P., Zhou, Y.: Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces. Forum Math. 25, 787–819 (2013)

    MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  22. Hajłasz, P.: Sobolev spaces on an arbitrary metric spaces. Potential Anal. 5, 403–415 (1996)

    MATH  MathSciNet  Google Scholar 

  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)

  24. Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688), x+101 (2000)

  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. Haroske, D.D., Triebel, H.: Embeddings of function spaces: a criterion in terms of differences. Complex Var. Elliptic Equ. 56, 931–944 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)

    Book  MATH  Google Scholar 

  28. Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  30. Hu, J.: A note on Hajłasz–Sobolev spaces on fractals. J. Math. Anal. Appl. 280, 91–101 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Jonsson, A., Wallin, H.: Boundary value problems and Brownian motion on fractals. Chaos Solitons Fractals 8, 191–205 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  32. Jonsson, A., Wallin, H.: Function spaces on subsets of \(\mathbb{R}^{n}\). Math. Rep. 2(1), xiv+221 (1984)

  33. Koskela, P., MacManus, P.: Quasiconformal mappings and Sobolev spaces. Studia Math. 131, 1–17 (1998)

    MATH  MathSciNet  Google Scholar 

  34. Koskela, P., Saksman, E.: Pointwise characterizations of Hardy–Sobolev functions. Math. Res. Lett. 15, 727–744 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  36. Koskela, P., Yang, D., Zhou, Y.: Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings. Adv. Math. 226, 3579–3621 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  37. McShane, E.J.: Extension of range of functions. Bull. Am. Math. Soc. 40, 837–842 (1934)

    Article  MathSciNet  Google Scholar 

  38. Müller, D., Yang, D.: A difference characterization of Besov and Triebel–Lizorkin spaces on RD-spaces. Forum Math. 21, 259–298 (2009)

  39. Ohtsuka, M.: Extremal Length and Precise Functions, GAKUTO International Series, Mathematical Sciences and Applications 19, pp. vi+343. Gakkōtosho Co., Ltd., Tokyo (2003)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  41. Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16, 243–279 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  42. Shanmugalingam, N.: Harmonic functions on metric spaces. Illinois J. Math. 45, 1021–1050 (2001)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  44. Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)

    Book  Google Scholar 

  45. Triebel, H.: Theory of Function Spaces II. Birkhäuser, Basel (1992)

    Book  MATH  Google Scholar 

  46. Triebel, H.: Fractals and Spectra, Related to Fourier Analysis and Function Spaces, pp. viii+271. Birkhäuser, Basel (1997)

  47. Triebel, H.: Theory of Function Spaces III, pp. xii+426. Birkhäuser, Basel (2006)

  48. Triebel, H.: Sobolev–Besov spaces of measurable functions. Stud. Math. 201, 69–86 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  49. Triebel, H.: Limits of Besov norms. Arch. Math. 96, 169–175 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  50. Väisäla, J.: Lectures on \(n\)-dimensional Quasiconformal Mapping, Lecture notes in Mathematics 229, pp. xiv+144. Springer, Berlin, New York (1971)

  51. Vishik, M.: Hydrodynamics in Besov spaces. Arch. Rational Mech. Anal. 145, 197–214 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  52. Yang, D.: New characterizations of Hajłasz–Sobolev spaces on metric spaces. Sci. China Ser. A 46, 675–689 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  53. Yang, D., Zhou, Y.: New properties of Besov and Triebel–Lizorkin spaces on RD-spaces. Manuscripta Math. 134, 59–90 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

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

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, OH , 45221-0025, USA

    Nageswari Shanmugalingam

  2. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China

    Dachun Yang & Wen Yuan

Authors
  1. Nageswari Shanmugalingam
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  2. Dachun Yang
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  3. Wen Yuan
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Corresponding author

Correspondence to Dachun Yang.

Additional information

Nageswari Shanmugalingam is partially supported by Grant No. 200474 from the Simons Foundation. Dachun Yang is supported by the National Natural Science Foundation of China (Grant Nos. 11171027 and 11361020). Wen Yuan is supported by the National Natural Science Foundation of China (Grant No. 11101038). This project is also partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003) and the Fundamental Research Funds for Central Universities of China (Grant Nos. 2012LYB26 and 2012CXQT09).

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Shanmugalingam, N., Yang, D. & Yuan, W. Newton–Besov spaces and Newton–Triebel–Lizorkin spaces on metric measure spaces. Positivity 19, 177–220 (2015). https://doi.org/10.1007/s11117-014-0291-7

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