We prove the following statement, which is a quantitative form of the Luzin theorem on C-property: Let (X, d, μ) be a bounded metric space with metric d and regular Borel measure μ that are related to one another by the doubling condition. Then, for any function f measurable on X, there exist a positive increasing function η ∈ Ω (η(+0) = 0 and η(t)t −a decreases for a certain a > 0), a nonnegative function g measurable on X, and a set E ⊂ X, μE = 0 , for which
If f ∈ L p (X), p >0, then it is possible to choose g belonging to L p (X).
Similar content being viewed by others
References
R. R. Coifman and G. Weiss, “Analyse harmonique non-commutative sur certain espaces homogenés,” Lect. Notes Math., 242, 1–176 (1971).
P. L. Ul’yanov, “Representation of functions by series and the classes φ(L),” Usp. Mat. Nauk, 27, No. 2, 3–52 (1972).
J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York (2001).
V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).
A. P. Calderon, “Estimates for singular integral operators in terms of maximal functions,” Stud. Math., 44, 561–582 (1972).
A. P. Calderon and R. Scott, “Sobolev type inequalities for p > 0,” Stud. Math., 62, 75–92 (1978).
R. DeVore and R. Sharpley, “Maximal functions measuring local smoothness,” Mem. Amer. Math. Soc., 47, 1–115 (1984).
K. I. Oskolkov, “Approximation properties of summable functions on sets of complete measure,” Mat. Sb., 103, No. 4, 563-589 (1977).
V. I. Kolyada, “Estimates for maximal functions related to local smoothness,” Dokl. Akad. Nauk SSSR, 293, No. 4, 534–537 (1987).
V. I. Kolyada, “Estimates of maximal functions measuring local smoothness,” Anal. Math., 25, 277–300 (1999).
D. Yang, “New characterization of Hajłasz–Sobolev spaces,” Sci. China, Ser. 1, 46, No. 5, 675–689 (2003).
I. A. Ivanishko, “Estimates for maximal Calderon – Kolyada functions on spaces of homogeneous type,” Tr. Inst. Mat. Nats. Akad. Nauk Belarusi, 12, No. 1, 64–67 (2004).
V. G. Krotov, “Weight L p -inequalities for sharp-maximal functions on metric spaces with measure,” Izv. Nats. Akad. Nauk Armenii, Ser. Mat., 41, No. 2, 27–42 (2006).
I. A. Ivanishko and V. G. Krotov, “Generalized Poincaré – Sobolev inequality on metric spaces,” Tr. Inst. Mat. Nats. Akad. Nauk Belarusi, 14, No. 1, 51–61 (2006).
I. A. Ivanishko and V. G. Krotov, “Compactness of imbeddings of Sobolev type on metric spaces with measure,” Mat. Zametki, 86, No. 6, 829–844 (2009).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 3, pp. 387–395, March, 2010.
Rights and permissions
About this article
Cite this article
Krotov, V.G. Quantitative form of the Luzin C-property. Ukr Math J 62, 441–451 (2010). https://doi.org/10.1007/s11253-010-0363-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-010-0363-1