Open Access Research Article

Hölder Quasicontinuity in Variable Exponent Sobolev Spaces

Petteri Harjulehto1*, Juha Kinnunen2 and Katja Tuhkanen2

Author Affiliations

1 Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hällströmin Katu 2b), Helsinki 00014, Finland

2 Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu 90014, Finland

For all author emails, please log on.

Journal of Inequalities and Applications 2007, 2007:032324 doi:10.1155/2007/32324


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2007/1/032324


Received:28 May 2006
Revisions received:6 November 2006
Accepted:25 December 2006
Published:14 February 2007

© 2007 Harjulehto et al.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We show that a function in the variable exponent Sobolev spaces coincides with a Hölder continuous Sobolev function outside a small exceptional set. This gives us a method to approximate a Sobolev function with Hölder continuous functions in the Sobolev norm. Our argument is based on a Whitney-type extension and maximal function estimates. The size of the exceptional set is estimated in terms of Lebesgue measure and a capacity. In these estimates, we use the fractional maximal function as a test function for the capacity.

References

  1. Kováčik, O, Rákosník, J: On spaces and . Czechoslovak Mathematical Journal. 41(116)(4), 592–618 (1991)

  2. Fan, X, Zhao, D: On the spaces and . Journal of Mathematical Analysis and Applications. 263(2), 424–446 (2001). Publisher Full Text OpenURL

  3. Acerbi, E, Mingione, G: Regularity results for stationary electro-rheological fluids. Archive for Rational Mechanics and Analysis. 164(3), 213–259 (2002). Publisher Full Text OpenURL

  4. Acerbi, E, Mingione, G: Regularity results for electrorheological fluids: the stationary case. Comptes Rendus de l'Académie des Sciences - Series I - Mathematics. 334(9), 817–822 (2002)

  5. Diening, L, Růžička, M: Strong solutions for generalized Newtonian fluids. Journal of Mathematical Fluid Mechanics. 7(3), 413–450 (2005). Publisher Full Text OpenURL

  6. Růžička, M: Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics,p. xvi+176. Springer, Berlin, Germany (2000)

  7. Růžička, M: Modeling, mathematical and numerical analysis of electrorheological fluids. Applications of Mathematics. 49(6), 565–609 (2004). Publisher Full Text OpenURL

  8. Chen, Y, Levine, S, Rao, M: Variable exponent, linear growth functionals in image restoration. SIAM Journal on Applied Mathematics. 66(4), 1383–1406 (2006). Publisher Full Text OpenURL

  9. Zhikov, VV: On some variational problems. Russian Journal of Mathematical Physics. 5(1), 105–116 (1997)

  10. Bojarski, B, Hajłasz, P, Strzelecki, P: Improved approximation of higher order Sobolev functions in norm and capacity. Indiana University Mathematics Journal. 51(3), 507–540 (2002)

  11. Evans, LC, Gariepy, RF: Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics,p. viii+268. CRC Press, Boca Raton, Fla, USA (1992)

  12. Gutiérrez, S: Lusin approximation of Sobolev functions by Hölder continuous functions. Bulletin of the Institute of Mathematics. Academia Sinica. 31(2), 95–116 (2003)

  13. Hajłasz, P, Kinnunen, J: Hölder quasicontinuity of Sobolev functions on metric spaces. Revista Matemática Iberoamericana. 14(3), 601–622 (1998)

  14. Liu, FC: A Luzin type property of Sobolev functions. Indiana University Mathematics Journal. 26(4), 645–651 (1977). Publisher Full Text OpenURL

  15. Malý, J: Hölder type quasicontinuity. Potential Analysis. 2(3), 249–254 (1993). Publisher Full Text OpenURL

  16. Michael, JH, Ziemer, WP: A Lusin type approximation of Sobolev functions by smooth functions. Classical Real Analysis (Madison, Wis., 1982), Contemp. Math., pp. 135–167. American Mathematical Society, Providence, RI, USA (1985)

  17. Acerbi, E, Fusco, N: Semicontinuity problems in the calculus of variations. Archive for Rational Mechanics and Analysis. 86(2), 125–145 (1984). Publisher Full Text OpenURL

  18. Lewis, JL: On very weak solutions of certain elliptic systems. Communications in Partial Differential Equations. 18(9-10), 1515–1537 (1993)

  19. Gurka, P, Harjulehto, P, Nekvinda, A: Bessel potential spaces with variable exponent. to appear in Mathematical Inequalities & Applications

  20. Capone, C, Cruz-Uribe, D, Fiorenza, A: The fractional maximal operator on variable spaces. preprint, 2004, http://www.na.iac.cnr.it/ webcite

  21. Cruz-Uribe, D, Fiorenza, A, Neugebauer, CJ: The maximal function on variable spaces. Annales Academiae Scientiarum Fennicae. Mathematica. 28(1), 223–238 (2003)

  22. Cruz-Uribe, D, Fiorenza, A, Neugebauer, CJ: Corrections to: "The maximal function on variable spaces". Annales Academiae Scientiarum Fennicae. Mathematica. 29(1), 247–249 (2004)

  23. Diening, L: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bulletin des Sciences Mathématiques. 129(8), 657–700 (2005)

  24. Diening, L: Maximal function on generalized Lebesgue spaces . Mathematical Inequalities & Applications. 7(2), 245–253 (2004). PubMed Abstract | Publisher Full Text OpenURL

  25. Kokilashvili, V, Samko, S: Maximal and fractional operators in weighted spaces. Revista Matemática Iberoamericana. 20(2), 493–515 (2004)

  26. Lerner, AK: Some remarks on the Hardy-Littlewood maximal function on variable spaces. Mathematische Zeitschrift. 251(3), 509–521 (2005). Publisher Full Text OpenURL

  27. Nekvinda, A: Hardy-Littlewood maximal operator on . Mathematical Inequalities & Applications. 7(2), 255–265 (2004). PubMed Abstract | Publisher Full Text OpenURL

  28. Musielak, J: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics,p. iii+222. Springer, Berlin, Germany (1983)

  29. Harjulehto, P, Hästö, P, Koskenoja, M, Varonen, S: Sobolev capacity on the space . Journal of Function Spaces and Applications. 1(1), 17–33 (2003)

  30. Diening, L: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces and . Mathematische Nachrichten. 268(1), 31–43 (2004). Publisher Full Text OpenURL

  31. Harjulehto, P, Hästö, P: Lebesgue points in variable exponent spaces. Annales Academiae Scientiarum Fennicae. Mathematica. 29(2), 295–306 (2004)

  32. Pick, L, Růžička, M: An example of a space on which the Hardy-Littlewood maximal operator is not bounded. Expositiones Mathematicae. 19(4), 369–371 (2001). Publisher Full Text OpenURL

  33. Kinnunen, J, Saksman, E: Regularity of the fractional maximal function. The Bulletin of the London Mathematical Society. 35(4), 529–535 (2003). Publisher Full Text OpenURL