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A note on a theorem of Gehring and Lehto


Author: A. G. Fadell
Journal: Proc. Amer. Math. Soc. 49 (1975), 195-198
MSC: Primary 26A54; Secondary 30A60
DOI: https://doi.org/10.1090/S0002-9939-1975-0367132-3
MathSciNet review: 0367132
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Abstract: The concept of mesh approximate differential is defined as a modification of regular approximate differential. It is shown that for open continuous real-valued maps on open sets in $ n$-space the concepts of mesh approximate differentiability and total differentiability are equivalent, and the Gehring-Lehto theorem is obtained as a corollary by a sharpening of a known theorem on regular approximate differentials.


References [Enhancements On Off] (What's this?)

  • [1] A. P. Calderón, On the differentiability of absolutely continuous functions, Riv. Mat. Univ. Parma 2 (1951), 203-213. MR 13, 544. MR 0045200 (13:544a)
  • [2] L. Cesari, Sulle funzioni assolutamente continue in due variabili, Ann. Scuola Norm. Sup. Pisa (2) 10 (1941), 91-101. MR 3, 230. MR 0005911 (3:230e)
  • [3] A. G. Fadell, On the existence of regular approximate differentials, Proc. Amer. Math. Soc. 37 (1973), 541-544. MR 47 #2016. MR 0313462 (47:2016)
  • [4] -, A generalization of Rademacher's theorem on complete differential, Colloq. Math. 27 (1973), 125-131. MR 0330372 (48:8709)
  • [5] F. W. Gehring and O. Lehto, On the total differentiability of functions of a complex variable, Ann. Acad. Sci. Fenn. Ser. A I No. 272 (1959), 9 pp. MR 23 #A1799. MR 0124487 (23:A1799)
  • [6] O. Lehto and K. I. Virtanen, Quasikonforme Abbildungen, Die Grundlehren der math. Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 126, Springer-Verlag, Berlin and New York, 1965. MR 32 #5872. MR 0188434 (32:5872)
  • [7] H. M. Riemann, Ueber das Verhalten von Flachen unter quasiconformen Abbildungen im Raum, Ann. Acad. Sci. Fenn. Ser. A I No. 470 (1970), 1-26. MR 0296290 (45:5351)
  • [8] J. Serrin, On the differentiability of functions of several variables, Arch. Rational Mech. Anal. 7 (1961), 359-372. MR 25 #3131. MR 0139700 (25:3131)
  • [9] J. Väisälä, Two new characterizations for conformality, Ann. Acad. Sci. Fenn. Ser. A I No. 362 (1965), 12 pp. MR 30 #4975. MR 0174782 (30:4975)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0367132-3
Keywords: Total differential, regular approximate differential, mesh approximate differential, open continuous mapping
Article copyright: © Copyright 1975 American Mathematical Society

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