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Science China Mathematics

, Volume 60, Issue 11, pp 1981–2010 | Cite as

Traces of weighted function spaces: Dyadic norms and Whitney extensions

  • Pekka Koskela
  • Tomás Soto
  • Zhuang Wang
Articles

Abstract

The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in the 1950’s. In this paper, we review the literature concerning such results for a variety of weighted smoothness spaces. For this purpose, we present a characterization of the trace spaces (of fractional order of smoothness), based on integral averages on dyadic cubes, which is well-adapted to extending functions using the Whitney extension operator.

Keywords

trace theorems weighted Sobolev spaces Besov spaces Triebel-Lizorkin spaces 

MSC(2010)

46E35 42B35 

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Notes

Acknowledgements

This work was supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (Grant No. 307333).

References

  1. 1.
    Aronszajn N. Boundary value of functions with finite Dirichlet integral. Techn Report 14. Kansas: University of Kansas, 1955zbMATHGoogle Scholar
  2. 2.
    Björn J. Poincaré inequalities for powers and products of admissible weights. Ann Acad Sci Fenn Math, 2001, 26: 175–188zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bonk M, Saksman E. Sobolev spaces and hyperbolic fillings. J Reine Angew Math, 2017, in pressGoogle Scholar
  4. 4.
    Bonk M, Saksman E, Soto T. Triebel-Lizorkin spaces on metric spaces via hyperbolic fillings. Indiana Univ Math J, 2017, in pressGoogle Scholar
  5. 5.
    Caetano A, Haroske D. Traces of Besov spaces on fractal h-sets and dichotomy results. Studia Math, 2015, 231: 117–147zbMATHMathSciNetGoogle Scholar
  6. 6.
    Cao J, Chang D-C, Yang D, et al. Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems. Trans Amer Math Soc, 2013, 365: 4729–4809CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Dhara R N, Kałamajska A. On one extension theorem dealing with weighted Orlicz-Slobodetskii space: Analysis on cube. Math Inequal Appl, 2015, 18: 61–89zbMATHMathSciNetGoogle Scholar
  8. 8.
    Dhara R N, Kałamajska A. On one extension theorem dealing with weighted Orlicz-Slobodetskii space: Analysis on Lipschitz subgraph and Lipschitz domain. Math Inequal Appl, 2016, 19: 451–488zbMATHMathSciNetGoogle Scholar
  9. 9.
    Dyda B, Ihnatsyeva L, Lehrbäck J, et al. Muckenhoupt A p-properties of distance functions and applications to Hardy- Sobolev-type inequalities. ArXiv:1705.01360, 2017Google Scholar
  10. 10.
    Fefferman C, Stein E M. H p spaces of several variables. Acta Math, 1972, 129: 137–193CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Fougères A. Théorèmes de trace et de prolongement dans les espaces de Sobolev et Sobolev-Orlicz. C R Acad Sci Paris Sér A, 1972, 274: 181–184zbMATHMathSciNetGoogle Scholar
  12. 12.
    Frazier M, Jawerth B. A discrete transform and decompositions of distribution spaces. J Funct Anal, 1990, 93: 34–170CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Gagliardo E. Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend Semin Mat Univ Padova, 1957, 27: 284–305zbMATHMathSciNetGoogle Scholar
  14. 14.
    Gogatishvili A, Koskela P, Shanmugalingam N. Interpolation properties of Besov spaces defined on metric spaces. Math Nachr, 2010, 283: 215–231CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Gogatishvili A, Koskela P, Zhou Y. Characterizations of Besov and Triebel-Lizorkin spaces on metric measure spaces. Forum Math, 2013, 25: 787–819zbMATHMathSciNetGoogle Scholar
  16. 16.
    Goldberg D. A local version of real Hardy spaces. Duke Math J, 1979, 46: 27–42CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    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, 2008, Art. ID 893409Google Scholar
  18. 18.
    Heikkinen T, Ihnatsyeva L, Tuominen H. Measure density and extension of Besov and Triebel-Lizorkin functions. J Fourier Anal Appl, 2016, 22: 334–382CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Heinonen J. Lectures on Analysis on Metric Spaces. New York: Springer-Verlag, 2001CrossRefzbMATHGoogle Scholar
  20. 20.
    Heinonen J, Kilpeläinen T, Martio O. Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford: Clarendon Press, 1993zbMATHGoogle Scholar
  21. 21.
    Heinonen J, Koskela P, Shanmugalingam N, et al. Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients. Cambridge: Cambridge University Press, 2015CrossRefzbMATHGoogle Scholar
  22. 22.
    Ihnatsyeva L, Vähäkangas A. Characterization of traces of smooth functions on Ahlfors regular sets. J Funct Anal, 2013, 265: 1870–1915CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Jonsson A. Besov spaces on closed sets by means of atomic decomposition. Complex Var Elliptic Equ, 2009, 54: 585–611CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Jonsson A, Wallin H. Function Spaces on Subsets of Rn. New York: Harwood Academic Publishers, 1984zbMATHGoogle Scholar
  25. 25.
    Koskela P, Saksman E. Pointwise characterizations of Hardy-Sobolev functions. Math Res Lett, 2008, 15: 727–744CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Lacroix M-Th. Espaces de traces des espaces de Sobolev-Orlicz. J Math Pures Appl (9), 1974, 53: 439–458zbMATHMathSciNetGoogle Scholar
  27. 27.
    Lahti P, Shanmugalingam N. Trace theorems for functions of bounded variation in metric spaces. ArXiv:1507.07006, 2015zbMATHGoogle Scholar
  28. 28.
    Lizorkin P I. Boundary properties of functions from “weight” classes (in Russian). Dokl Akad Nauk SSSR, 1960, 132: 514–517; translated as Soviet Math Dokl, 1960, 1: 589–593Google Scholar
  29. 29.
    Malý L. Trace and extension theorems for Sobolev-type functions in metric spaces. ArXiv:1704.06344, 2017Google Scholar
  30. 30.
    Malý L, Shanmugalingam N, Snipes M. Trace and extension theorems for functions of bounded variation. ArXiv: 1511.04503, 2015Google Scholar
  31. 31.
    Mironescu P. Note on Gagliardo’s theorem “trW1,1 = L1”. Ann Univ Buchar Math Ser, 2015, 6: 99–103zbMATHMathSciNetGoogle Scholar
  32. 32.
    Mironescu P, Russ E. Traces of weighted Sobolev spaces. Old and new. Nonlinear Anal, 2015, 119: 354–381CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Miyachi A. Maximal functions for distributions on open sets. Hitotsubashi J Arts Sci, 1987, 28: 45–58MathSciNetGoogle Scholar
  34. 34.
    Miyachi A. H p spaces over open subsets of Rn. Studia Math, 1990, 95: 205–228CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Miyachi A. Hardy-Sobolev spaces and maximal functions. J Math Soc Japan, 1990, 42: 73–90CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Neˇcas J. Direct Methods in the Theory of Elliptic Equations. Heidelberg: Springer, 2012CrossRefGoogle Scholar
  37. 37.
    Nikolskii S M. Properties of certain classes of functions of several variables on differentiable manifolds (in Russian). Mat Sb, 1953, 33: 261–326MathSciNetGoogle Scholar
  38. 38.
    Palmieri G. The traces of functions in a class of Sobolev-Orlicz spaces with weight. Boll Unione Mat Ital (9), 1981, 18: 87–117MathSciNetGoogle Scholar
  39. 39.
    Peetre J. New Thoughts on Besov Spaces. Durham: Duke University, 1976zbMATHGoogle Scholar
  40. 40.
    Saka K. The trace theorem for Triebel-Lizorkin spaces and Besov spaces on certain fractal sets. I. The restriction theorem. Mem College Ed Akita Univ Natur Sci, 1995, 48: 1–17zbMATHMathSciNetGoogle Scholar
  41. 41.
    Saka K. The trace theorem for Triebel-Lizorkin spaces and Besov spaces on certain fractal sets. II. The extension theorem. Mem College Ed Akita Univ Natur Sci, 1996, 49: 1–27zbMATHMathSciNetGoogle Scholar
  42. 42.
    Saksman E, Soto T. Traces of Besov, Triebel-Lizorkin and Sobolev spaces on metric spaces. ArXiv:1606.08729, 2016Google Scholar
  43. 43.
    Shvartsman P. Local approximations and intrinsic characterization of spaces of smooth functions on regular subsets of Rn. Math Nachr, 2006, 279: 1212–1241CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Slobodetskii L N, Babich V M. On boundedness of the Dirichlet integrals (in Russian). Dokl Akad Nauk SSSR (NS), 1956, 106: 604–606Google Scholar
  45. 45.
    Soto T. Besov spaces on metric spaces via hyperbolic fillings. ArXiv:1606.08082, 2016Google Scholar
  46. 46.
    Stein E. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton University Press, 1993zbMATHGoogle Scholar
  47. 47.
    Strömberg J-O, Torchinsky A. Weighted Hardy Spaces. Berlin: Springer-Verlag, 1989CrossRefzbMATHGoogle Scholar
  48. 48.
    Triebel H. Theory of Function Spaces. Basel: Birkhäuser Verlag, 1983CrossRefzbMATHGoogle Scholar
  49. 49.
    Triebel H. The Structure of Functions. Basel: Birkhäuser Verlag, 2001CrossRefzbMATHGoogle Scholar
  50. 50.
    Tyulenev A I. Description of traces of functions in the Sobolev space with a Muckenhoupt weight. Proc Steklov Inst Math, 2014, 284: 280–295CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Tyulenev A I. Boundary values of functions in a Sobolev space with weight of Muckenhoupt class on some non-Lipschitz domains. Mat Sb, 2014, 205: 67–94; translation in Sb Math, 2014, 205: 1133–1159CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Tyulenev A I. Traces of weighted Sobolev spaces with Muckenhoupt weight: The case p = 1. Nonlinear Anal, 2015, 128: 248–272CrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    Tyulenev A I. Some new function spaces of variable smoothness. Mat Sb, 2015, 206: 85–128; translation in Sb Math, 2015, 206: 849–891CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Tyulenev A I, Vodop’yanov S K. On a Whitney-type problem for weighted Sobolev spaces on d-thick closed sets. ArXiv:1606.06749, 2016zbMATHGoogle Scholar
  55. 55.
    Vašarin A A. The boundary properties of functions having a finite Dirichlet integral with a weight (in Russian). Dokl Akad Nauk SSSR (NS), 1957, 117: 742–744Google Scholar
  56. 56.
    Wang H, Yang X. The characterization of the weighted local Hardy spaces on domains and its application. J Zhejiang Univ Sci Ed, 2004, 9: 1148–1154CrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland

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