# Change of variables formula under minimal assumptions

Colloquium Mathematicae (1993)

- Volume: 64, Issue: 1, page 93-101
- ISSN: 0010-1354

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topHajłasz, Piotr. "Change of variables formula under minimal assumptions." Colloquium Mathematicae 64.1 (1993): 93-101. <http://eudml.org/doc/210177>.

@article{Hajłasz1993,

abstract = {In the previous papers concerning the change of variables formula (in the form involving the Banach indicatrix) various assumptions were made about the corresponding transformation (see e.g. [BI], [GR], [F], [RR]). The full treatment of the case of continuous transformation is given in [RR]. In [BI] the transformation was assumed to be continuous, a.e. differentiable and with locally integrable Jacobian. In this paper we show that none of these assumptions is necessary (Theorem 2). We only need the a.e. existence of approximate partial derivatives. In Section 3 we consider the general form of the change of variables formula for Sobolev mappings. The author wishes to thank Professor Bogdan Bojarski for many stimulating conversations and suggestions.},

author = {Hajłasz, Piotr},

journal = {Colloquium Mathematicae},

keywords = {approximately totally differentiable function; Lusin condition (N); change of variables formula; Banach indicatrix; Sobolev mappings},

language = {eng},

number = {1},

pages = {93-101},

title = {Change of variables formula under minimal assumptions},

url = {http://eudml.org/doc/210177},

volume = {64},

year = {1993},

}

TY - JOUR

AU - Hajłasz, Piotr

TI - Change of variables formula under minimal assumptions

JO - Colloquium Mathematicae

PY - 1993

VL - 64

IS - 1

SP - 93

EP - 101

AB - In the previous papers concerning the change of variables formula (in the form involving the Banach indicatrix) various assumptions were made about the corresponding transformation (see e.g. [BI], [GR], [F], [RR]). The full treatment of the case of continuous transformation is given in [RR]. In [BI] the transformation was assumed to be continuous, a.e. differentiable and with locally integrable Jacobian. In this paper we show that none of these assumptions is necessary (Theorem 2). We only need the a.e. existence of approximate partial derivatives. In Section 3 we consider the general form of the change of variables formula for Sobolev mappings. The author wishes to thank Professor Bogdan Bojarski for many stimulating conversations and suggestions.

LA - eng

KW - approximately totally differentiable function; Lusin condition (N); change of variables formula; Banach indicatrix; Sobolev mappings

UR - http://eudml.org/doc/210177

ER -

## References

top- [BI] B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in ${R}^{n}$, Ann. Acad. Sci. Fenn. Ser. AI Math. 8 (1983), 257-324. Zbl0548.30016
- [F] H. Federer, Geometric Measure Theory, Springer, 1969. Zbl0176.00801
- [GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York 1983. Zbl0562.35001
- [GR] V. M. Goldshteĭn and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Mappings, Nauka, Moscow 1983 (in Russian).
- [H] P. Hajłasz, Co-area formula, Sobolev mappings and related topics, in preparation. Zbl0988.28002
- [He] L. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510. Zbl0283.26003
- [IM] T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Mittag-Leffler Report #19, 1989/90 (to appear in Acta Math.). Zbl0785.30008
- [K] M. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen, Fund. Math. 22 (1934), 77-108. Zbl60.0532.03
- [M] O. Martio, Lusin's condition (N) and mappings with non-negative Jacobians, preprint. Zbl0807.46032
- [P1] S. P. Ponomarev, An example of an $ACT{L}^{p}$ homeomorphism which is not absolutely continuous in Banach sense, Dokl. Akad. Nauk SSSR 201 (1971), 1053-1054 (in Russian).
- [P2] S. P. Ponomarev, On the property N for homeomorphisms of class ${W}_{p}^{1}$, Sibirsk. Mat. Zh. 28 (2) (1987), 140-148 (in Russian).
- [RR] T. Rado and P. V. Reichelderfer, Continuous Transformations in Analysis, Springer, 1955. Zbl0067.03506
- [R1] Yu. G. Reshetnyak, Some geometrical properties of functions and mappings with generalized derivatives, Sibirsk. Mat. Zh. 7 (4) (1966), 886-919 (in Russian).
- [R2] Yu. G. Reshetnyak, On the condition N for mappings of class ${W}_{n,loc}^{1}$, ibid. 28 (5) (1987), 149-153 (in Russian).
- [S] L. Simon, Lectures on Geometric Measure Theory, Proc. Centre Math. Anal. Austral. Nat. Univ. 3, 1983. Zbl0546.49019
- [V] J. Väisälä, Quasiconformal maps and positive boundary measure, Analysis 9 (1989), 205-216. Zbl0674.30018
- [W] H. Whitney, On totally differentiable and smooth functions, Pacific J. Math. 1 (1951), 143-159. Zbl0043.05803

## Citations in EuDML Documents

top- Piotr Hajłasz, Soheil Malekzadeh, On Conditions for Unrectifiability of a Metric Space
- Luděk Kleprlík, Composition operators on W 1 X are necessarily induced by quasiconformal mappings
- O. Martio, Lebesgue measure and mappings of the Sobolev class ${W}^{1,n}$
- Bogdan Bojarski, Piotr Hajłasz, Pointwise inequalities for Sobolev functions and some applications
- Marc Troyanov, Sergei Vodop'yanov, Liouville type theorems for mappings with bounded (co)-distortion

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