Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-26T07:48:31.433Z Has data issue: false hasContentIssue false
  • English
  • Français

Lower dimensional absolute continuity properties of quasiconformal mappings

Published online by Cambridge University Press:  24 October 2008

F. W. Gehring
F. W. Gehring
Affiliation:
University of Michigan, Ann Arbor, Michigan
Get access Rights & Permissions [Opens in a new window]

Extract

Let Rn denote Euclidean n-space. For p ∈ [0, ∞), the normalized Hausdorff p-dimensional outer measure of a set E ⊂ Rn is defined as

where the infimum is taken over all countable coverings of E by sets Ej with

and

(See, for example (4).) Then ℋP (E) measures the p-dimensional size of E. For example, ℋ0 (E) ≤ ∞ if and only if E is finite, in which case ℋ0(E) = card (E). Next if E is an arc or Jordan curve, then ℋ1(E) ≤ ∞ if and only if E is rectifiable, in which case ℋ1(E) = length (E). Finally if p is an integer in [1, n] and if E lies in a p-dimensional hyperplane T in Rn, then

where mp denotes the Lebesgue p-dimensional outer measure in T. Thus as p varies from 0 to n, the measures ℋp interpolate in a natural way between the counting measure and Lebesgue outer measure in Rn.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78 , Issue 1 , July 1975 , pp. 81 - 93
Copyright
Copyright © Cambridge Philosophical Society 1975

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Agard, S. B.Quasiconformal mappings and the moduli of p-dimensional surface families. Proceedings of the Romanian-Finnish Seminar on Teichmüller spaces and quasiconformal mappings Brasov, Romania, 1969.Google Scholar
(2)Aleksandrov, P. S.Combinatorial topology. (Graylock Press, Albany, New York, 1960).Google Scholar
(3)Alexandroff, P. and Hopf, H.Topologie. (Chelsea Publishing Co., Bronx, New York, 1965).Google Scholar
(4)Federer, H.Geometric measure theory. (Springer-Verlag, New York, 1969.)Google Scholar
(5)Gehring, F. W.The definitions and exceptional sets for quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I. 281 (1960), 1–28.Google Scholar
(6)Gehring, F. W.Symmetrization of rings in space. Trans. Amer. Math. Soc. 101 (1961), 499–519.CrossRefGoogle Scholar
(7)Gehring, F. W.Extension theorems for quasiconformal mappings in n-space. J. Analyse Math. 19 (1967), 149–169.CrossRefGoogle Scholar
(8)Gehring, F. W.Dilatations of quasiconformal boundary correspondences. Duke Math. J. 39 (1972), 89–95.CrossRefGoogle Scholar
(9)Gehring, F. W.The Hausdorff measure of sets which link in Euclidean space. Contributions to analysis: A collection of papers dedicated to Lipman Bers. (Academic Press, New York, 1974).CrossRefGoogle Scholar
(10)Gehring, F. W. Absolute continuity properties of quasiconformal mappings. Symposia Math, (to appear).Google Scholar
(11)Morse, A. P.A theory of covering and differentiation. Trans. Amer. Math. Soc. 55 (1944), 205–235.CrossRefGoogle Scholar
(12)Mostow, G. D.Quasiconformal mappings in n-space and the rigidity of hyperbolic space forms. Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104.CrossRefGoogle Scholar
(13)Saks, S.Theory of the integral (Hafner Publishing Co., New York, 1937).Google Scholar
(14)Sarvas, J.Symmetrization of condensers in n-space. Ann. Acad. Sci. Fenn. Ser A I. 522 (1972), 1–44.Google Scholar
(15)Spanier, E. H.Algebraic topology. (McGraw-Hill, New York, 1966).Google Scholar
(16)Väisälä, J.Lectures on n-dimensional quasiconformal mappings. Lecture notes in mathematics 229 (Springer-Verlag, Berlin-Heidelberg-New York, 1971).CrossRefGoogle Scholar