A note on a theorem of Gehring and Lehto

Fadell, A. G.

doi:10.1090/S0002-9939-1975-0367132-3

A note on a theorem of Gehring and Lehto
HTML articles powered by AMS MathViewer

by A. G. Fadell PDF

Proc. Amer. Math. Soc. 49 (1975), 195-198 Request permission

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

Alberto P. Calderón, On the differentiability of absolutely continuous functions, Riv. Mat. Univ. Parma 2 (1951), 203–213. MR 45200
Lamberto Cesari, Sulle funzioni assolutamente continue in due variabili, Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) 10 (1941), 91–101 (Italian). MR 5911
Albert Fadell, On the existence of regular approximate differentials, Proc. Amer. Math. Soc. 37 (1973), 541–544. MR 313462, DOI 10.1090/S0002-9939-1973-0313462-9
Albert G. Fadell, A generalization of Rademacher’s theorem on complete differential, Colloq. Math. 27 (1973), 125–131. MR 330372, DOI 10.4064/cm-27-1-125-131
F. W. Gehring and Olli Lehto, On the total differentiability of functions of a complex variable, Ann. Acad. Sci. Fenn. Ser. A I No. 272 (1959), 9. MR 0124487
O. Lehto and K. I. Virtanen, Quasikonforme Abbildungen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 126, Springer-Verlag, Berlin-New York, 1965 (German). MR 0188434
H. M. Riemann, Über das Verhalten von Flächen unter quasikonformen Abbildungen im Raum, Ann. Acad. Sci. Fenn. Ser. A. I. 470 (1970), 27 (German). MR 296290
James Serrin, On the differentiability of functions of several variables, Arch. Rational Mech. Anal. 7 (1961), 359–372. MR 139700, DOI 10.1007/BF00250769
Jussi Väisälä, Two new characterizations for quasiconformality, Ann. Acad. Sci. Fenn. Ser. A I No. 362 (1965), 12. MR 0174782