Skip to main content
Log in

Whitney’s jets for Sobolev functions

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We present two fundamental facts from the jet theory for Sobolev spaces W m, p. One of these facts is that the formal differentiation of the k-jets theory is compatible with the pointwise definition of Sobolev (m − 1)-jet spaces on regular subsets of the Euclidean spaces ℝn. The second result describes the Sobolev imbedding operator of Sobolev jet spaces increasing the order of integrability of Sobolev functions up to the critical Sobolev exponent.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. L. Sobolev, “On one theorem of functional analysis,” Mat. Sb., 46, 471–497 (1938).

    Google Scholar 

  2. S. L. Sobolev, Applications of Functional Analysis to Mathematical Physics (in Russian), Novosibirsk (1962).

  3. O. P. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems, Wiley, New York (1978).

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  5. P. Hajłasz, “Sobolev spaces on an arbitrary metric space,” Potential Anal., 5, 403–415 (1996).

    MATH  MathSciNet  Google Scholar 

  6. B. Bojarski, P. Hajłasz, and P. Strzelecki, “Improved C k λ approximation of higher-order Sobolev functions in norm and capacity,” Indiana Univ. Math. J., 51, 507–540 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Bojarski, “Marcinkiewicz-Zygmund theorem and Sobolev spaces,” Special Volume in Honour of L. D. Kudryavtsev’s 80th Birthday, Fizmatlit, Moscow (2003).

    Google Scholar 

  8. F. C. Liu and W. S. Tai, “Approximate Taylor polynomials and differentiation of functions,” Top. Meth. Nonlinear Anal., 3, 189–196 (1994).

    MATH  MathSciNet  Google Scholar 

  9. H. Whitney, “Analytic extensions of differentiable functions defined in closed sets,” Trans. Amer. Math. Soc., 36, 63–89 (1934).

    Article  MATH  MathSciNet  Google Scholar 

  10. H. Whitney, “Differentiable functions defined in closed sets. I,” Trans. Amer. Math. Soc., 36, 369–387 (1934).

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Jonsson and H. Wallin, “Function spaces on subsets of R n,” Math. Repts., 2, No. 1 (1984).

  12. B. Malgrange, Ideals of Differentiable Functions, Oxford University Press, London (1967).

    Google Scholar 

  13. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton (1970).

    MATH  Google Scholar 

  14. G. Glaeser, “Étude de quelques algèbres tayloriennes,” J. Anal. Math., 6, 1–124 (1958).

    MATH  MathSciNet  Google Scholar 

  15. W. P. Ziemer, “Weakly differentiable functions. Sobolev spaces and functions of bounded variation,” Grad. Texts Math., 120 (1989).

  16. B. Bojarski, “Pointwise characterization of Sobolev classes,” Proc. Steklov Inst. Math. (to appear).

  17. B. Muckenhoupt and R. L. Wheeden, “Weighted norm inequalities for fractional integrals,” Trans. Amer. Math. Soc., 192, 261–274 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  18. B. Bojarski, “Sharp maximal operators of fractional order and Sobolev imbedding inequalities,” Bull. Polish Acad. Sci. Math., 33, 7–16 (1985).

    MATH  MathSciNet  Google Scholar 

  19. A. P. Calderón, “Estimates for singular integral operators in terms of maximal functions,” Stud. Math., 44, 563–582 (1972).

    MATH  Google Scholar 

  20. R. A. de Vore and R. C. Sharpley, “Maximal functions measuring smoothness,” Mem. Amer. Math. Soc., 47, No. 293, (1984).

    Google Scholar 

  21. D. Swanson, “Pointwise inequalities and approximation in fractional Sobolev spaces,” Stud. Math., 149, 147–174 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  22. P. Hajłasz and P. Koskela, “Sobolev met Poincaré,” Mem. Amer. Math. Soc., 145, No. 688 (2000).

  23. B. Bojarski, “Remarks on Markov’s inequalities and some properties of polynomials,” Bull. Polish Acad. Sci. Math., 33, 355–365 (1985).

    MATH  MathSciNet  Google Scholar 

  24. L. Hedberg, “On certain convolution inequalities,” Proc. Amer. Math. Soc., 36, 505–510 (1972).

    Article  MathSciNet  Google Scholar 

  25. P. Hajłasz and J. Kinnunen, “Hölder quasicontinuity of Sobolev functions on metric spaces,” Rev. Mat., 14, 601–622 (1998).

    MATH  Google Scholar 

  26. P. Hajłasz, P. Koskela, and H. Tuominen, Sobolev Extensions and Restriction, Preprint.

  27. P. Shvartsman, Local Approximations and Intrinsic Characterizations of Spaces of Smooth Functions on Regular Subsets of R n, ArXiv: Math. FA/0601682.

  28. H. Triebel, “A new approach to function spaces on quasimetric spaces,” Rev. Mat. Comput., 18, 7–48 (2005).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 345–358, March, 2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bojarski, B. Whitney’s jets for Sobolev functions. Ukr Math J 59, 379–395 (2007). https://doi.org/10.1007/s11253-007-0024-1

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-007-0024-1

Keywords

Navigation