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Quasiregular mappings in even dimensions. (English) Zbl 0785.30008

In their introduction to this long important paper the authors say that the paper has grown from their study of a paper of S. K. Donaldson and D. P. Sullivan [DS] [Acta Math. 163, No. 3/4, 181-252 (1989; Zbl 0704.57008)]. Using and developing the ideas of [DS] the authors study differential forms with integrable coefficients and Hodge theory on \(L^ p\)-spaces and with great success apply these methods to investigate several topics and open problems in the theory of quasiconformal and quasiregular maps in \(R^ n\), \(n \geq 2\). Among such topics are (a) a definition of a new Beurling-Ahlfors type singular integral operator \(S\) suited for higher dimensional studies, (b) regularity questions in Liouville’s theorem in even dimensions, (c) discovery of new differential equations satisfied by quasiregular maps in even dimensions, (d) new definition of quasiregular maps, (e) derivation of Caccioppoli type estimates.
The authors give various examples showing the sharpness of their results and also present various conjectures. It is not possible to adequately describe even the main results of the paper in this review. Therefore we refer the reader to a recent survey of the first author [in Lect. Notes Math. 1508, 39-64 (1992; reviewed below)] where also this paper is discussed and conclude by what the authors regard as their most important result. This result deals with the study of exceptional sets of bounded quasiregular maps. Here the main problem is the following one. Let \(E\) be a compact subset of the unit ball \(B^ n\) of measure zero and let \(f:B^ n \backslash E \to R^ n\) be a bounded quasiregular map. Under which conditions on \(E\) and \(f\) the map \(f\) has an extension to \(E\)? Yu. G. Reshetnyak [Sib. Mat. Zh. 10, 1300-1310 (1969; Zbl 0201.098)] and O. Martio, S. Rickman and J. Väisälä [Ann. Acad. Sci. Fenn., Ser. A I 465, 1-13 (1970; Zbl 0197.057)] settled the case when \(E\) is of zero \(n\)-capacity. But what about the case when \(E\) is of positive Hausdorff dimension? With little success this problem was studied by E. A. Poletskij [Mat. Sb., Nov. Ser. 92(134), 242-256 (1973; Zbl 0281.30017)] and the reviewer [Ann. Acad. Sci. Fenn., Ser. A I, Diss. 11, 44 p. (1976; Zbl 0362.30024)]. The authors obtain now a sufficient condition for the removability in even dimensions \(n=2l\) in terms of a condition that involves \(lp\)-capacity of \(E\), \(p\)-norm of the \(S\)-operator and the “Beltrami coefficient” (which the authors define) of \(f\). In this context, related simultaneous results should be mentioned, namely the papers of P. Koskela and O. Martio [Ann. Acad. Sci. Fenn., Ser. AI 15, No. 2, 381-399 (1990; Zbl 0717.30015)], of P. Järvi and the reviewer [J. Reine Angew. Math. 424, 31-45 (1992; Zbl 0733.30017)] and of S. Rickman (to appear)]. The first author has later extended some of the results also to the case of odd dimensions [Ann. Math. 136, 651-685 (1992; reviewed below)].

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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[1] Ahlfors, L. V.,Lectures on Quasiconformal Mappings. Van Nostrand, Princeton, 1966; Reprinted by Wadsworth Inc., Belmont, 1987. · Zbl 0138.06002
[2] –, Conditions for quasiconformal deformations in several variables, inContributions to Analysis, pp. 19–25. Academic Press, New York, 1974.
[3] Ahlfors, L. V. &Beurling, A., Conformal invariants and function theoretic null sets.Acta Math., 83 (1950), 101–129. · Zbl 0041.20301 · doi:10.1007/BF02392634
[4] Boyarski, B. V., Homeomorphic solutions of Beltrami systems.Dokl. Akad. Nauk SSSR, 102 (1955), 661–664.
[5] Boyarski, B. V. &Iwaniec, T., Another approach to Liouville Theorem.Math. Nachr., 107 (1982), 253–262. · Zbl 0527.30013 · doi:10.1002/mana.19821070120
[6] –, Analytical foundations of the theory of quasiconformal mappings in R n .Ann. Acad. Sci. Fenn. Ser. A I Math., 8 (1983), 257–324. · Zbl 0548.30016
[7] Donaldson, S. K. &Sullivan, D. P., Quasiconformal 4-manifolds.Acta Math., 163 (1989), 181–252. · Zbl 0704.57008 · doi:10.1007/BF02392736
[8] Edwards, R. E.,Fourier Series, vol. II. Holt, Rinehart, Winston, 1967.
[9] Fifiel, T., Iwaniec, T. &Pelczynski, A., Computing norms and critical exponents of some operators inL p -spaces.Studia Math., 79 (1984), 227–274.
[10] Flanders, H.,Differential Forms. Academic Press, 1963. · Zbl 0112.32003
[11] Freed, D. S. & Uhlenbeck, K. K.,Instantons and Four Manifolds. Math. Sci. Res. Publ., 1. Springer-Verlag, 1972. · Zbl 0559.57001
[12] Garnett, J.,Analytic Capacity and Measure. Lecture Notes in Math., 297. Springer-Verlag, 1972. · Zbl 0253.30014
[13] Garcia-Cuerva, J. & Rubio de Francia, J. L.,Weighted Norm Inequalities and Related Topics. Notas de Matemática, 104; North-Holland Math. Studies, 116. North-Holland, 1985. · Zbl 0578.46046
[14] Gehring, F. W., Rings and quasiconformal mappings in space.Trans. Amer. Math. Soc., 103 (1962), 353–393. · Zbl 0113.05805 · doi:10.1090/S0002-9947-1962-0139735-8
[15] –, TheL p -integrability of the partial derivatives of a quasiconformal mapping.Acta Math., 130 (1973), 265–277. · Zbl 0258.30021 · doi:10.1007/BF02392268
[16] Gehring, F. W., Topics in quasiconformal mappings, inProceedings of the ICM, Berkeley, 1986, pp. 62–82.
[17] Granlund, S., Lindqvist, P. &Martio, O., Conformally invariant variational integrals.Trans. Amer. Math. Soc., 277 (1983), 43–73. · Zbl 0518.30024 · doi:10.1090/S0002-9947-1983-0690040-4
[18] Iwaniec, T., Some aspects of partial differential equations and quasiregular mappings, inProceedings of the ICM, Warsaw, 1983, pp. 1193–1208.
[19] Iwaniec, T.,Regularity Theorems for the Solutions of Partial Differential Equations Related to Quasiregular Mappings in Several Variables. Preprint Polish Acad. Sci., Habilitation Thesis, pp. 1–45, 1978; Dissertationes Mathematicae, CXCVIII, 1982.
[20] Iwaniec, T., On Cauchy-Riemann derivatives in several real variables.Lecture Notes in Math., 1039 (1983), 220–224. Springer-Verlag. · Zbl 0544.30020
[21] –, Projections onto gradient fields andL p -estimates for degenerate elliptic operators.Studia Math., 75 (1983), 293–312. · Zbl 0552.35034
[22] –,p-harmonic tensors and quasiregular mappings.Ann. of Math., 136 (1992), 651–685. · Zbl 0785.30009 · doi:10.2307/2946602
[23] Iwaniec, T. &Martin, G. J., Quasiconformal mappings and capacity.Indiana Math. J., 40 (1991), 101–122. · Zbl 0752.30009 · doi:10.1512/iumj.1991.40.40005
[24] Iwaniec, T. & Martin, G. J., The Beurling-Ahlfors transform in R n and related singular integrals. I.H.E.S. Preprint, 1990.
[25] Järvi, P. & Vuorinen, M., Self-similar Cantor sets and quasiregular mappings. Institut Mittag-Leffler Report, no. 27, 1989/90; To appear inJ. Reine Angew. Math. · Zbl 0733.30017
[26] Koskela, P. & Martio, O., Removability theorems for quasiregular mappings. To appear inAnn. Acad. Sci. Fenn. Ser. A I Math. · Zbl 0717.30015
[27] Lawson, H. B. & Michelson, M. L.,Spin Geometry. Princeton Univ. Press, 1989.
[28] Lehto, O., Quasiconformal mappings and singular integrals.Sympos. Math., XVIII (1976), 429–453. Academic Press, London. · Zbl 0339.30017
[29] Lelong-Ferrand, J., Geometrical interpretations of scalar curvature and regularity of conformal homeomorphisms, inDifferential Geometry and Relativity (A. Lichnerowicz Sixtieth Birthday Volume), Reidel, 1976, pp. 91–105.
[30] Liouville, J., Théorème sur l’équationdx 2+dy 2+dz 2 = {\(\lambda\)}(d{\(\alpha\)} 2+d{\(\beta\)} 2+d{\(\gamma\)} 2).J. Math. Pures Appl. 1, 15 (1850), 103.
[31] Manfredi, J., Regularity for minima of functionals withp-growth.J. Differential Equations, 76 (1988), 203–212. · Zbl 0674.35008 · doi:10.1016/0022-0396(88)90070-8
[32] Maz’ja, G. V.,Sobolev Spaces. Springer-Verlag, 1985.
[33] Martio, O., Rickman, S. &Väisälä, J., Definitions for quasiregular mappings.Ann. Acad. Sci. Fenn. Ser. AI Math., 448 (1969), 1–40. · Zbl 0189.09204
[34] –, Distortion and singularities of quasiregular mappings.Ann. Acad. Sci. Fenn. Ser. A I Math., 465 (1970), 1–13. · Zbl 0197.05702
[35] –, Topological and metric properties of quasiregular mappings.Ann. Acad. Sci. Fenn. Ser. A I Math., 488 (1971), 1–31. · Zbl 0223.30018
[36] Martio, O. &Sarvas, J., Injective theorems in plane and space.Ann. Acad. Sci. Fenn. Ser. A I Math., 4 (1979), 383–401. · Zbl 0406.30013
[37] Martio, O. &Srebro, U., Automorphic quasimeromorphic mappings.Acta Math., 135 (1975), 221–247. · Zbl 0329.30012 · doi:10.1007/BF02392020
[38] Poletsky, E. A., On the removal of singularities of quasiconformal mappings.Math. USSSR-Sb., 21 (1973), 240–254. · Zbl 0289.30028 · doi:10.1070/SM1973v021n02ABEH002015
[39] Peltonen, K.,Quasiregular Mappings onto S n . Lic. Thesis, Helsinki, 1988.
[40] Reshetnyak, Y. G., Liouville’s conformal mapping theorem under minimal regularity assumptions.Sibirsk. Mat. Zh., 8 (1967), 835–840.
[41] –, Differentiable properties of quasiconformal mappings and conformal mappings of Riemannian spaces.Sibirsk. Mat. Zh., 19 (1978), 1166–1183.
[42] –Space Mappings with Bounded Distortion. Transl. Math. Monographs, 73, Amer. Math. Soc., Providence, 1989. · Zbl 0667.30018
[43] Rickman, S., The analogue of Picard’s Theorem for quasiregular mappings in dimension three.Acta Math., 154 (1985), 195–242. · Zbl 0617.30024 · doi:10.1007/BF02392472
[44] –, Asymptotic values and angular limits of quasiregular mappings of a ball.Ann. Acad. Sci. Fenn. Ser. A I Math., 5 (1980), 185–196. · Zbl 0453.30003
[45] Rickman, S., Nonremovable Cantor sets for bounded quasiregular mappings. To appear inAnn. Acad. Sci. Fenn. Ser. A I Math. · Zbl 0823.30012
[46] Rickman, S., Quasiregular mappings. To appear.
[47] Sarvas, J., Quasiconformal semiflows.Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), 197–219. · Zbl 0551.30019
[48] Stein, E. M.,Singular Integrals and Differentiable Properties of Functions. Princeton Univ. Press, 1970. · Zbl 0207.13501
[49] Teleman, N., The Index Theorem for topological manifolds.Acta Math., 153 (1984), 117–152. · Zbl 0547.58036 · doi:10.1007/BF02392376
[50] Tukia, P., Hausdorff dimension and quasisymmetric mappings.Math. Scand., 65 (1989), 152–160. · Zbl 0677.30016
[51] –, Automorphic quasimeromorphic mappings for torisonless hyperbolic groups.Ann. Acad. Sci. Fenn. Ser. A I Math., 10 (1985), 545–560. · Zbl 0545.30012
[52] Vuorinen, M.,Conformal Geometry and Quasiregular Mappings. Lecture Notes in Math., 1319. Springer-Verlag, 1988.
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