Abstract
We prove that known estimates for capacities and the Hausdorff dimension of exceptional Lebesgue sets of functions from Calderón classes are sharp. Our proof also gives a similar result for classical Sobolev spaces.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Here and thereafter we denote by c different positive constants, whose values play no role.
- 2.
Often x is called a Lebesgue point, if (6) is true for q = 1.
- 3.
The authors would like to thank the referee for these references.
References
Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, Berlin (2001)
Federer, H., Ziemer, W.: The Lebesgue sets of a function whose distribution derivatives are pth power summable. Indiana Univ. Math. J. 22(2), 139–158 (1972)
Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory, p. 366. Springer, Berlin, Heidelberg, New York (1996)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, London, New York, Washington (1992)
Ziemer, W.P.: Weakly differentiable functions. Sobolev Spaces and Functions of Bounded Variation. Springer, New York (1989)
Calderón, A.P.: Estimates for singular integral operators in terms of maximal functions. Studia Math. 44, 561–582 (1972)
Yang, D.: New characterization of Hajłasz-Sobolev spaces on metric spaces. Science in China, Ser. 1. 46(5), 675–689 (2003)
DeVore, R., Sharpley, R.: Maximal functions measuring local smoothness. Memoirs Amer. Math. Soc. 47, 1–115 (1984)
Hajłasz, P.: Sobolev spaces on an arbitrary metric spaces. Potential Anal. 5(4), 403–415 (1996)
Ivanishko, I.A.: Generalized Sobolev classes on metric measure spaces. Math. Notes. 77(6), 865–869 (2005)
Kinnunen, J., Martio, O.: The Sobolev capacity on metric spaces. Annales Academiæ Scientiarum Fennicæ Mathematica 21, 367–382 (1996)
Prokhorovich, M.A.: Capacity and Lebesque points for Sobolev classes. Bulletin of the National Academy of Sciences of Belarus 1, 19–23 (2006) (In Russian)
Prokhorovich, M.A.: Hausdorff dimension of Lebesgue sets for W α p classes on metric spaces. Math. Notes. 82(1), 88-95 (2007)
Hajłasz, P., Kinnunen, J.: Hölder qasicontinuity of Sobolev functions on metric spaces. Revista Matemática Iberoamericana. 14(3), 601–622 (1998)
Kinnunen, J., Latvala, V.: Lebesgue points for Sobolev functions on metric spaces. Revista Matemática Iberoamericana. 18(3), 685–700 (2002)
Prokhorovich, M.A.: Hausdorff measures and Lebesgue points for the Sobolev classes W α p, α > 0 on spaces of homogeneous type. Math. Notes. 85(4), 584–589 (2009)
Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (1990)
Fonseca, I., Maly, J., Mingione, G.: Scalar minimizers with fractal singular sets. Arch. Rat. Mech. Anal. 172(3), 295–307 (2004)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
Dedicated to Professor Konstantin Oskolkov on the occasion of his 65th birthday.
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this paper
Cite this paper
Krotov, V.G., Prokhorovich, M.A. (2012). Estimates for the Exceptional Lebesgue Sets of Functions from Sobolev Classes. In: Bilyk, D., De Carli, L., Petukhov, A., Stokolos, A., Wick, B. (eds) Recent Advances in Harmonic Analysis and Applications. Springer Proceedings in Mathematics & Statistics, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4565-4_19
Download citation
DOI: https://doi.org/10.1007/978-1-4614-4565-4_19
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-4564-7
Online ISBN: 978-1-4614-4565-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)