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Absolutely continuous functions of several variables and diffeomorphisms. (English) Zbl 1033.26020

Summary: In [J. Malý, J. Math. Anal. Appl. 231, 492–508 (1999; Zbl 0924.26008)] a class of absolutely continuous functions of \(d\)-variables, motivated by applications to change of variables in an integral, has been introduced. The main result of this paper states that absolutely continuous functions in the sense of Malý (loc. cit.) are not stable under diffeomorphisms. We also show an example of a function which is absolutely continuous with respect to cubes but not with respect to balls.

MSC:

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

Citations:

Zbl 0924.26008
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References:

[1] M. Csörnyei: “Absolutely continuous functions of Rado, Reichelderfer and Malý”, J. Math. Anal. Appl., Vol. 252, (2000), pp. 147-166. http://dx.doi.org/10.1006/jmaa.2000.6962; · Zbl 0981.26009
[2] S. Hencl: “On the notions of absolute continuity for functions of several variables”, Fund. Math., Vol. 173, (2002), pp. 175-189. http://dx.doi.org/10.4064/fm173-2-5; · Zbl 1002.26007
[3] S. Hencl: “Absolutely continuous functions of several variables and quasiconformal mappings”, preprint MATH-KMA-2002/89, Charles University, Prague.; · Zbl 1065.26017
[4] J. Malý: “Absolutely continuous function of several variables”, J. Math. Anal. Appl., Vol. 231, (1999), pp. 492-508. http://dx.doi.org/10.1006/jmaa.1998.6246; · Zbl 0924.26008
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