A Pointwise Condition for the Absolute Continuity of a Function of One Variable and Its Applications

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/m7572-3270-2461-v A Pointwise Condition for the Absolute Continuity of a Function of One Variable and Its Applications Vodopyanov, S. K. Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 4. Abstract: An absolutely continuous function in calculus is precisely such a function that, within the framework of Lebesgue integration, can be restored from its derivative, that is, the Newton-Leibniz theorem on the relationship between integration and differentiation is fulfilled for it. An equivalent definition is that the the sum of the moduli of the increments of the function with respect to arbitrary pair-wise disjoint intervals is less than any positive number if the sum of the lengths of the intervals is small enough. Certain sufficient conditions for absolute continuity are known, for example, the Banach-Zaretsky theorem. In this paper we prove a new sufficient condition for the absolute continuity of a function of one variable and give some of its applications to problems in the theory of function spaces. The proved condition makes it possible to significantly simplify the proof of the theorems on the pointwise description of functions of the Sobolev classes defined on Euclidean spaces and Сarnot groups. Keywords: absolutely continuous function, Sobolev space, pointwise description Language: English Download the full text For citation: Vodopyanov, S. K. A Pointwise Condition for the Absolute Continuity of a Function of One Variable and Its Applications, Vladikavkaz Math. J., 2021, vol. 23, no. 4, pp. 41-49. DOI 10.46698/m7572-3270-2461-v + References 1. Natanson, I. P. Theory of Functions of a Real Variable, Moscow-Leningrad, Gostekhizdat, 1950; English transl., Frederick Ungar Publ. Co., New York, 1955. 2. Reshetnyak, Yu. G. Space Mappings with Bounded Distortion, Providence, Amer. Math. Soc., 1989. 3. Vodop'yanov, S. K. Regularity of Mappings Inverse to Sobolev Mappings, Sbornik: Mathematics, 2012, vol. 203, no. 10, pp. 1-28. DOI: 10.1070/SM2012v203n10ABEH004269. 4. Hajlasz, P. Sobolev Spaces on an Arbitrary Metric Space, Potential Analysis, 1996, vol. 5, no. 4, pp. 403-415. 5. Vodopyanov, S. K. Monotone Functions and Quasiconformal Mappings on Carnot groups, Siberian Mathematical Journal, 1996, vol. 37, no. 6, pp. 1269-1295. DOI: 10.1007/BF02106736. 6. Bojarski, B. Remarks on Some Geometric Properties of Sobolev Mappings, Functional Analysis & Related Topics (Sapporo, 1990), pp. 65-76; World Sci. Publ., River Edge, NJ, 1991. 7. Jain, P., Molchanova, A., Singh, M. and Vodopyanov, S. On Grand Sobolev Spaces and Pointwise Description of Banach Function Spaces Nonlinear Analysis, Theory, Methods and Applications, 2021, vol. 202, no. 1, pp. 1-17. https://doi.org/10.1016/j.na.2020.112100. 8. Bonfiglioli, A., Lanconelli, E. and Uguzzoni, F. Stratified Lie Groups and Potential Theory for their Sub-Laplacians, Berlin, Heidelberg, Springer-Verlag, 2007. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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