Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A strong Lebesgue point property for Sobolev functions


Author: Visa Latvala
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2331-2338
MSC (2000): Primary 46E35; Secondary 31C15
DOI: https://doi.org/10.1090/S0002-9939-04-07358-7
Published electronically: February 19, 2004
MathSciNet review: 2052410
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that first-order Sobolev functions fulfill a Wiener integral type Lebesgue point property outside a set of Sobolev capacity zero. Our condition is stronger than the standard Lebesgue point property, but the exceptional set is slightly larger.


References [Enhancements On Off] (What's this?)

  • 1. D. R. Adams and L. I. Hedberg, Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften, Band 314, Springer-Verlag, Berlin, Heidelberg, 1996. MR 97j:46024
  • 2. D. R. Adams and J. L. Lewis, Fine and quasiconnectedness in nonlinear potential theory, Ann. Inst. Fourier (Grenoble), 35.1 (1985), pp. 53-73. MR 86h:31009
  • 3. D. R. Adams and N. G. Meyers, Thinness and Wiener criteria for non-linear potentials, Indiana Univ. Math. J., 22 (1972/73), pp. 169-197. MR 47:5272
  • 4. L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, Boca Raton, Florida, 1992. MR 93f:28001
  • 5. H. Federer and W. P. Ziemer, The Lebesgue set of a function whose distribution derivatives are $p$-th power summable, Indiana Univ. Math. J., 22 (1972/73), pp. 139-158.MR 55:8321
  • 6. P. Hajlasz and J. Kinnunen, Hölder quasicontinuity of Sobolev functions on metric spaces, Rev. Mat. Iberoamericana, 14 (1998), pp. 601-622. MR 2000e:46046
  • 7. L. I. Hedberg, Non-linear potentials and approximation in the mean by analytic functions, Math. Z., 129 (1972), pp. 299-319.MR 48:6430
  • 8. J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential theory of degenerate elliptic equations, The Clarendon Press, Oxford University Press, New York, 1993. MR 94e:31003
  • 9. J. Kinnunen and N. Shanmugalingam, Regularity of quasi-minimizers on metric spaces, Manuscripta Math., 105 (2001), pp. 401-423. MR 2002i:35054
  • 10. T. Kilpeläinen and J. Malý, Supersolutions to degenerate elliptic equations on quasi open sets, Comm. Partial Differential Equations, 17 (1992), pp. 371-405. MR 93g:31022
  • 11. T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), pp. 137-161. MR 95a:35050
  • 12. J. Malý, Hölder type quasicontinuity, Potential Analysis, 2 (1993), pp. 249-254. MR 94i:31007
  • 13. N. G. Meyers, Continuity properties of potentials, Duke Math. J., 42 (1975), pp. 157-166. MR 51:3477
  • 14. V. G. Maz'ya and V. P. Khavin, Nonlinear potential theory, Uspehi Mat. Nauk 27 (1972), pp. 67-138; Russian Math. Surveys, 27.6 (1972), pp. 71-148. MR 53:13610
  • 15. J. Malý and W. P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, No. 51, Amer. Math. Soc., Providence, RI, 1997. MR 98h:35080
  • 16. W. P. Ziemer, Weakly differentiable functions, ``Sobolev spaces and functions of bounded variation'', Graduate Texts in Mathematics, No. 120, Springer-Verlag, New York, 1989. MR 91e:46046

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46E35, 31C15

Retrieve articles in all journals with MSC (2000): 46E35, 31C15


Additional Information

Visa Latvala
Affiliation: Department of Mathematics, University of Joensuu, P.O. Box 111, 80101 Joensuu, Finland
Email: visa.latvala@joensuu.fi

DOI: https://doi.org/10.1090/S0002-9939-04-07358-7
Keywords: Sobolev functions, Lebesgue points, capacity
Received by editor(s): January 23, 2003
Received by editor(s) in revised form: April 29, 2003
Published electronically: February 19, 2004
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society