Change of variables formula under minimal assumptions

Piotr Hajłasz

Colloquium Mathematicae (1993)

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

Abstract

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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.

How to cite

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Hajłasz, 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

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  1. [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
  2. [F] H. Federer, Geometric Measure Theory, Springer, 1969. Zbl0176.00801
  3. [GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York 1983. Zbl0562.35001
  4. [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). 
  5. [H] P. Hajłasz, Co-area formula, Sobolev mappings and related topics, in preparation. Zbl0988.28002
  6. [He] L. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510. Zbl0283.26003
  7. [IM] T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Mittag-Leffler Report #19, 1989/90 (to appear in Acta Math.). Zbl0785.30008
  8. [K] M. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen, Fund. Math. 22 (1934), 77-108. Zbl60.0532.03
  9. [M] O. Martio, Lusin's condition (N) and mappings with non-negative Jacobians, preprint. Zbl0807.46032
  10. [P1] S. P. Ponomarev, An example of an A C T L p homeomorphism which is not absolutely continuous in Banach sense, Dokl. Akad. Nauk SSSR 201 (1971), 1053-1054 (in Russian). 
  11. [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). 
  12. [RR] T. Rado and P. V. Reichelderfer, Continuous Transformations in Analysis, Springer, 1955. Zbl0067.03506
  13. [R1] Yu. G. Reshetnyak, Some geometrical properties of functions and mappings with generalized derivatives, Sibirsk. Mat. Zh. 7 (4) (1966), 886-919 (in Russian). 
  14. [R2] Yu. G. Reshetnyak, On the condition N for mappings of class W n , l o c 1 , ibid. 28 (5) (1987), 149-153 (in Russian). 
  15. [S] L. Simon, Lectures on Geometric Measure Theory, Proc. Centre Math. Anal. Austral. Nat. Univ. 3, 1983. Zbl0546.49019
  16. [V] J. Väisälä, Quasiconformal maps and positive boundary measure, Analysis 9 (1989), 205-216. Zbl0674.30018
  17. [W] H. Whitney, On totally differentiable and smooth functions, Pacific J. Math. 1 (1951), 143-159. Zbl0043.05803

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