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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.
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 Ñarnot 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
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