Traces and extensions of certain weighted Sobolev spaces on \(\mathbb {R}^n\) and Besov functions on Ahlfors regular compact subsets of \(\mathbb {R}^n\)

Abstract

The focus of this paper is on Ahlfors Q-regular compact sets \(E\subset \mathbb {R}^n\) such that, for each \(Q-2<\alpha \le 0\), the weighted measure \(\mu _{\alpha }\) given by integrating the density \(\omega (x)=\text {dist}(x, E)^\alpha \) yields a Muckenhoupt \(\mathcal {A}_p\)-weight in a ball B containing E. For such sets E we show the existence of a bounded linear trace operator acting from \(W^{1,p}(B,\mu _\alpha )\) to \(B^\theta _{p,p}(E, \mathcal {H}^Q\vert _E)\) when \(0<\theta <1-\tfrac{\alpha +n-Q}{p}\), and the existence of a bounded linear extension operator from \(B^\theta _{p,p}(E, \mathcal {H}^Q\vert _E)\) to \(W^{1,p}(B, \mu _\alpha )\) when \(1-\tfrac{\alpha +n-Q}{p}\le \theta <1\). We illustrate these results with E as the Sierpiński carpet, the Sierpiński gasket, and the von Koch snowflake.

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References

  1. 1.

    Ahlfors, L.V.: Quasiconformal reflections. Acta Math. 109, 291–301 (1963)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Barton, A.: Trace and extension theorems relating Besov spaces to weighted averaged Sobolev spaces. Math. Inequal. Appl. 21(3), 817–870 (2018)

    MathSciNet  MATH  Google Scholar 

  3. 3.

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

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Björn, A., Björn, J., Shanmugalingam, N.: Extension and trace results for doubling metric measure spaces and their hyperbolic fillings. in preparation

  5. 5.

    Björn, J., Shanmugalingam, N.: Poincaré inequalities, uniform domains and extension properties for Newton-Sobolev functions in metric spaces. J. Math. Anal. Appl. 332(1), 190–208 (2007)

    MathSciNet  Article  Google Scholar 

  6. 6.

    David, G., Feneuil, J., Mayboroda, S.: Harmonic measure on sets of codimension larger than one. C. R. Math. Acad. Sci. Paris 355(4), 406–410 (2017)

    MathSciNet  Article  Google Scholar 

  7. 7.

    David, G., Feneuil, J., Mayboroda, S.: Elliptic theory in domains with boundaries of mixed dimension preprint, arXiv:2003.09037v2 (2020) 1–116

  8. 8.

    Dyda, B., Ihnatsyeva, L., Lehrbäck, J., Tuominen, H., Vähäkangas, A.V.: Muckenhoupt \(A_p\)-properties of distance functions and applications to Hardy-Sobolev-type inequalities. Potential Anal. 50, 83–105 (2019)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Gogatishvili, A., Koskela, P., Shanmugalingam, N.: Interpolation properties of Besov spaces defined on metric spaces. Math. Nachr. 283(2), 215–231 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Gogatishvili, A., Koskela, P., Zhou, Y.: Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces. Forum Math. 25(4), 787–819 (2013)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Hajłasz, P., Martio, O.: Traces of Sobolev functions on fractal type sets and characterization of extension domains. J. Funct. Anal. 143(1), 221–246 (1997)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations, p. 404. Dover Publications, Minneola (2006)

    MATH  Google Scholar 

  13. 13.

    Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J. T..:Sobolev spaces on metric measure spaces. An approach based on upper gradients. New Mathematical Monographs, 27 (2015), Cambridge University Press, Cambridge, xii+434 pp

  14. 14.

    Herron, D.: The geometry of uniform, quasicircle, and circle domains. Ann. Acad. Sci. Fenn. Ser. A I Math. 12(2), 217–227 (1987)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Herron, D., Koskela, P.: Uniform and Sobolev extension domains. Proc. Am. Math. Soc. 114(2), 483–489 (1992)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Jonsson, A., Wallin, H.: The trace to subsets of Rn of Besov spaces in the general case. Anal. Math. 6(3), 223–254 (1980)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Jonsson, A., Wallin, H.: Function spaces on subsets of \(\mathbb{R}^n\). Math. Rep. 2 (1984), no. 1, xiv+221 pp

  18. 18.

    Keith, S., Zhong, X.: The Poincaré inequality is an open ended condition. Ann. Math. (2) 167(2), 575–599 (2008)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Koskela, P., Soto, T., Wang, Z.: Traces of weighted function spaces: dyadic norms and Whitney extensions. Sci. China Math. 60(11), 1981–2010 (2017)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Malý, L.:Trace and extension theorems for Sobolev-type functions in metric spaces. preprint, arXiv:1704.06344

  21. 21.

    Martio, O., Sarvas, J.: Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math. 4(2), 383–401 (1979)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Maz’ya, V.G.:Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften 342 (2011), Springer, Heidelberg, xxviii+866 pp

  23. 23.

    Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for fractional integrals. Trans. Am. Math. Soc. 192, 261–274 (1974)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Muckenhoupt, B., Wheeden, R.: Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform. Studia Math. 55(3), 279–294 (1976)

    MathSciNet  Article  Google Scholar 

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Correspondence to Nageswari Shanmugalingam.

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Dedicated with gratitude to Professor Pekka Koskela on his 59th birthday.

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Nageswari Shanmugalingam was partially supported by grant DMS #1800161 from the National Science Foundation (U.S.A.). Both authors also acknowledge a great debt to the custodial staff at the University of Cincinnati who maintained the facilities during these difficult times; this debt can never be adequately repaid.

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Lindquist, J., Shanmugalingam, N. Traces and extensions of certain weighted Sobolev spaces on \(\mathbb {R}^n\) and Besov functions on Ahlfors regular compact subsets of \(\mathbb {R}^n\). Complex Anal Synerg 7, 7 (2021). https://doi.org/10.1007/s40627-021-00064-1

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Keywords

  • Besov space
  • Weighted Sobolev space
  • Ahlfors regular sets
  • Sierpiński carpet
  • Gsket
  • von Koch snowflake
  • Trace
  • Extension

Mathematics Subject Classification

  • Primary 46E35
  • Secondary 31E05