On differential properties of functions of bounded variation.

*(English)*Zbl 1299.26036Summary: It is established that Karagulyan’s exact estimate of the divergence rate of strong integral means of summable functions is extendable to strong means of additive functions of intervals having bounded variation. Furthermore, it is proved that each function defined on \([0, 1]^n\) with bounded variation in the sense of Hardy has a strong gradient at almost every point (this strengthens the corresponding result of Burkill and Haslam-Jones on the differentiability almost everywhere), whereas the same is not true for functions with bounded variation in the sense of Arzela.

##### MSC:

26B30 | Absolutely continuous real functions of several variables, functions of bounded variation |

##### Keywords:

divergence rate; strong means of additive functions with bounded variation; strong gradient; functions with bounded variation in the sense of Hardy and Arzela
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\textit{L. D. Bantsuri} and \textit{G. G. Oniani}, Anal. Math. 38, No. 1, 1--17 (2012; Zbl 1299.26036)

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##### References:

[1] | C. R. Adams and J. A. Clarkson, Properties of functions f(x, y) of bounded variation, Trans. Amer. Math. Soc., 36(1934), no. 4, 711–730. · JFM 60.0201.04 |

[2] | L. D. Bantsuri and G. G. Oniani, On the differential properties of functions of bounded variation in Hardy sense, Proc. A. Razmadze Math. Inst., 139(2005), 93–95. · Zbl 1096.26503 |

[3] | L. Bantsuri and G. Oniani, On the divergence rate of strong means of additive functions of intervals with bounded variation, Bull. Georgian Natl. Acad. Sci., 173(2006), no. 3, 453–454. |

[4] | J. C. Burkill and U. S. Haslam-Jones, Notes on the differentiability of functions of two variables, J. London Math. Soc., 7(1932), 297–305. · Zbl 0005.39104 · doi:10.1112/jlms/s1-7.4.297 |

[5] | H. Busemann and W. Feller, Zur Differentiation der Lebesgueschen Integrale, Fundamenta Math., 22(1934), 226–256. · JFM 60.0218.03 |

[6] | J. A. Clarkson and C. R. Adams, On definitions of bounded variation for functions of two variables, Trans. Amer. Math. Soc., 35(1933), no. 4, 824–854. · Zbl 0008.00602 · doi:10.1090/S0002-9947-1933-1501718-2 |

[7] | O. P. Dzagnidze, On the differentiability of functions of two variables and of indefinite double integrals, Proc. A. Razmadze Math. Inst., 106(1993), 7–48. · Zbl 0836.26007 |

[8] | O. Dzagnidze and G. Oniani, On one analogue of Lebesgue theorem on the differentiation of indefinite integral for functions of several variables, Proc. A. Razmadze Math. Inst., 133(2003), 1–5. · Zbl 1054.26501 |

[9] | E. W. Hobson, The theory of functions of a real variable and the theory of Fourier’s series. Vol. II, Dover Publications (New York, N.Y., 1958). · Zbl 0081.27702 |

[10] | L. D. Ivanov, Variations of sets and functions (edited by A. G. Vituškin), Izdat. ”Nauka” (Moscow, 1975) (in Russian). · Zbl 0967.26500 |

[11] | B. Jessen, J. Marcinkiewicz, and A. Zygmund, Note on the differentiability of multiple integrals, Fundamenta Math., 25(1935), 217–234. · Zbl 0012.05901 |

[12] | G. Karagulyan, On the growth of integral means of functions from L 1(R n), East J. Approx., 3(1997), no. 1, 1–12. |

[13] | A. S. Kronrod, On functions of two variables, Uspekhi Matem. Nauk (N.S.), 5(1)(1950), 24–134 (in Russian). |

[14] | A. S. Leonov, Remarks on the total variation of functions of several variables and on a multidimensional analogue of Helly’s choice principle, Mat. Zametki, 63(1998), no. 1, 69–80 (in Russian); translation in Math. Notes, 63(1998), no. 1–2, 61–71. · Zbl 0924.26007 · doi:10.4213/mzm1249 |

[15] | I. P. Natanson, Theory of functions of a real variable, third edition, Izdat. ”Nauka” (Moscow, 1974) (in Russian). |

[16] | G. G. Oniani, On the inter-relation between differentiability conditions and the existence of a strong gradient, Mat. Zametki 77(2005), no. 1, 93–98 (in Russian); translation in Math. Notes 77(2005), no. 1–2, 84–89. · doi:10.4213/mzm2472 |

[17] | S. Saks, Remark on the differentiability of the Lebesgue indefinite integral, Fundamenta Math., 22(1934), 257–261. · Zbl 0009.10602 |

[18] | S. Saks, On the strong derivatives of functions of intervals, Fundamenta Math., 25(1935), 235–252. · JFM 61.0255.02 |

[19] | S. Saks, Theory of the integral, second revised edition, English translation by L. C. Young, with two additional notes by Stefan Banach, Dover Publications (New York, 1964). · Zbl 1196.28001 |

[20] | G. E. Šilov and B. L. Gurevič, Integral, measure and derivative. General theory, second revised edition, Izdat. ”Nauka” (Moscow, 1967) (in Russian). |

[21] | W. Stepanoff, Sur les conditions of l’existance de la différentialle totale, Moscow, Rec. Math., 32(1925), 511–527. |

[22] | A. G. Vituškin, On multidimensional variations, Gosudarstv. Izdat. Tehn.-Teor. Lit. (Moscow, 1955) (in Russian). |

[23] | W. H. Young and G. C. Young, On the discontinuities of monotone functions of several variables, Proc. London Math. Soc., 22(1924), 124–142. · JFM 49.0180.02 · doi:10.1112/plms/s2-22.1.124 |

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