Quasiconformal mappings with Sobolev boundary values

Astala, Kari; Bonk, Mario; Heinonen, Juha

Astala, Kari ; Bonk, Mario ; Heinonen, Juha

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 3, pp. 687-731.

Résumé

We consider quasiconformal mappings in the upper half space $ℝ_{+}^{n + 1}$ of $ℝ^{n + 1}$ , $n \geq 2$ , whose almost everywhere defined trace in $ℝ^{n}$ has distributional differential in $L^{n} (ℝ^{n})$ . We give both geometric and analytic characterizations for this possibility, resembling the situation in the classical Hardy space $H^{1}$ . More generally, we consider certain positive functions defined on $ℝ_{+}^{n + 1}$ , called conformal densities. These densities mimic the averaged derivatives of quasiconformal mappings, and we prove analogous trace theorems for them. The abstract approach of general conformal densities sheds new light to the mapping case as well.

Classification : 30C65, 46E35

@article{ASNSP_2002_5_1_3_687_0,
     author = {Astala, Kari and Bonk, Mario and Heinonen, Juha},
     title = {Quasiconformal mappings with {Sobolev} boundary values},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {687--731},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 1},
     number = {3},
     year = {2002},
     mrnumber = {1990676},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2002_5_1_3_687_0/}
}

TY  - JOUR
AU  - Astala, Kari
AU  - Bonk, Mario
AU  - Heinonen, Juha
TI  - Quasiconformal mappings with Sobolev boundary values
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2002
SP  - 687
EP  - 731
VL  - 1
IS  - 3
PB  - Scuola normale superiore
UR  - http://www.numdam.org/item/ASNSP_2002_5_1_3_687_0/
LA  - en
ID  - ASNSP_2002_5_1_3_687_0
ER  -

%0 Journal Article
%A Astala, Kari
%A Bonk, Mario
%A Heinonen, Juha
%T Quasiconformal mappings with Sobolev boundary values
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2002
%P 687-731
%V 1
%N 3
%I Scuola normale superiore
%U http://www.numdam.org/item/ASNSP_2002_5_1_3_687_0/
%G en
%F ASNSP_2002_5_1_3_687_0

Astala, Kari; Bonk, Mario; Heinonen, Juha. Quasiconformal mappings with Sobolev boundary values. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 3, pp. 687-731. http://www.numdam.org/item/ASNSP_2002_5_1_3_687_0/

Bibliographie
Cité par

[Am] L. Ambrosio, Metric space valued functions of bounded variation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 3 (1990), 439-478. | EuDML | Numdam | MR | Zbl

[As] K. Astala, Quasiharmonic analysis and BMO, University of Joensuu, Publications in Sciences, 14 (1989), 9-19. | MR | Zbl

[AG] K. Astala - F. W. Gehring, Quasiconformal analogues of theorems of Koebe and Hardy-Littlewood, Mich. Math. J. 32 (1985), 99-107. | MR | Zbl

[AK] K. Astala - P. Koskela, $H^{p}$ -theory for quasiconformal mappings, in preparation. | Zbl

[BL] Y. Benyamini - J. Lindenstrauss, “Geometric Nonlinear Functional Analysis”, Volume I, Vol. 48 of Colloquium Publications, American Mathematical Society, Providence, R.I., 2000. | MR | Zbl

[BM] A. Baernstein Ii - J. J. Manfredi, Topics in quasiconformal mapping, In: “Topics in Modern Harmonic Analysis”, Proc. Seminar held in Torino and Milano II (1982), 819-862. | MR | Zbl

[BA] A. Beurling - L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 124-142. | MR | Zbl

[BK] M. Bonk - P. Koskela, Conformal metrics and size of the boundary, Amer. J. Math., to appear. | MR | Zbl

[BHK] M. Bonk - J. Heinonen - P. Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque 270, 2001. | Numdam | MR | Zbl

[BHR] M. Bonk - J. Heinonen - S. Rohde, Doubling conformal densities, J. reine angew. Math. 541 (2001), 117-141. | MR | Zbl

[BKR] M. Bonk - P. Koskela - S. Rohde, Conformal metrics on the unit ball in Euclidean space, Proc. London Math. Soc. 77 (1998), 635-664. | MR | Zbl

[CST] A. Connes - D. Sullivan - N. Teleman, Quasiconformal mappings, operators on Hilbert space, and local formulae for characteristic classes, Topology 33 (1994), 663-681. | MR | Zbl

[DU] J. Diestel - J. J. Uhl, “Vector Measures”, American Mathematical Society, Providence, R.I., 1977. | MR | Zbl

[D] J. L. Doob, “Classical Potential Theory and its Probabilistic Counterpart”, Springer-Verlag, New York, 1984. | MR | Zbl

[Fe] H. Federer, “Geometric Measure Theory”, Springer-Verlag, Berlin-Heidelberg-New York, 1969. | MR | Zbl

[Fu] B. Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171-218. | MR | Zbl

[Ga] J. Garnett, “Bounded Analytic Functions”, Academic Press, New York, 1981. | MR | Zbl

[G1] F. W. Gehring, 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. | MR | Zbl

[G2] F. W. Gehring, Lower dimensional absolute continuity properties of quasiconformal mappings, Math. Proc. Cambridge Philos. Soc. 78 (1975), 81-93. | MR | Zbl

[G3] F. W. Gehring, Absolute continuity properties of quasiconformal mappings, Symposia Math. XVIII (1976), 551-559. | MR | Zbl

[GH] F. W. Gehring - W. K. Hayman, An inequality in the theory of conformal mapping, J. Math. Pure Appl. 41 (1962), 353-361. | MR | Zbl

[GO] F. W. Gehring - B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Anal. Math. 36 (1979), 50-74. | MR | Zbl

[GT] D. Gilbarg - N. S. Trudinger, “Elliptic Partial Differential Operators of Second Order”, 2nd Edition, Springer-Verlag, Berlin-Heidelberg-New York, 1983. | MR | Zbl

[HaK] P. Hajłasz - P. Koskela, Sobolev met Poincaré, Memoirs Amer. Math. Soc. 145, 2000. | MR | Zbl

[Ha] B. Hanson, Quasiconformal analogues of a theorem of Smirnov, Math. Scand. 75 (1994), 133-149. | MR | Zbl

[H1] J. Heinonen, The boundary absolute continuity of quasiconformal mappings, Amer. J. Math. 116 (1994), 1545-1567. | MR | Zbl

[H2] J. Heinonen, The boundary absolute continuity of quasiconformal mappings II, Rev. Mat. Iberoamericana 12 (1996), 697-725. | MR | Zbl

[H3] J. Heinonen, A theorem of Semmes and the boundary absolute continuity in all dimensions, Rev. Mat. Iberoamericana 12 (1996), 783-789. | MR | Zbl

[HKM] J. Heinonen - T. Kilpeläinen - O. Martio, “Nonlinear Potential Theory of Degenerate Elliptic Equations”, Oxford University Press, 1993. | MR | Zbl

[HeK] J. Heinonen - P. Koskela, The boundary distortion of a quasiconformal mapping, Pacific J. Math. 165 (1994), 93-114. | MR | Zbl

[HKST] J. Heinonen - P. Koskela - N. Shanmugalingam - J.T. Tyson, Sobolev classes of Banach space-valued functions and quasiconformal mappings, J. Anal. Math. 85 (2001), 87-139. | MR | Zbl

[HS] J. Heinonen - S. Semmes, Thirty-three $Y E S$ or $N O$ questions about mappings, measures, and metrics, Conformal Geom. Dyn. 1 (1997), 1-12. | MR | Zbl

[JW] D. Jerison - A. Weitsman, On the means of quasiregular and quasiconformal mappings, Proc. Amer. Math. Soc. 83 (1981), 304-306. | MR | Zbl

[J] P. W. Jones, Extension theorems for BMO, Indiana Univ. Math. J. 29 (1980), 41-66. | MR | Zbl

[K] W. Kaplan, On Gross's star theorem, schlicht functions, logarithmic potentials, and Fourier series, Ann. Acad. Sci. Fenn. Ser. A I Math.-Phys. 86 (1951), 1-23. | MR | Zbl

[KSh] S. Kallunki - N. Shanmugalingam, Modulus and continuous capacity, Ann. Acad. Sci. Fenn. Ser. A I Math. 26 (2001), 455-464. | MR | Zbl

[KS] N. J. Korevaar - R. M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), 561-659. | MR | Zbl

[KM] P. Koskela - P. Macmanus, Quasiconformal mappings and Sobolev spaces, Studia Math. 131 (1998), 1-17. | MR | Zbl

[KR] P. Koskela - S. Rohde, Hausdorff dimension and mean porosity, Math. Ann. 309 (1997), 593-609. | MR | Zbl

[Ma] N. G. Makarov, Probability methods in the theory of conformal mappings, (in Russian) Algebra and Analysis 1 (1989), 1-59. | MR | Zbl

[Mz] V. G. Maz'Ja, “Sobolev Spaces”, Springer-Verlag, Berlin, 1985. | MR

[MZ] J. Malý - W. P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, AMS Mathematical Surveys and Monographs, 51, Providence, R.I., 1997. | MR | Zbl

[Po] Ch. Pommerenke, “Boundary Behaviour of Conformal Maps”, Springer-Verlag, Berlin-Heidelberg-New York, 1992. | MR | Zbl

[Pr] I. I. Priwalow, “Randeigenschaften analytischer Funktionen”, Deutscher Verlag Wiss., Berlin, 1956. | MR | Zbl

[Re] Yu. G. Reshetnyak, Sobolev classes of functions with values in a metric space, Sibirsk. Mat. Zh. 38 (1997), 657-675. | MR | Zbl

[Ri] F. Riesz - M. Riesz, Über die Randwerte einer analytischen Funktion, 4. Cong. Scand. Math., Stockholm (1916), 27-44. | JFM

[RS1] R. Rochberg - S. Semmes, End point results for estimates of singular values of singular integral operators, Contributions to operator theory and its applications (Mosa, AZ, 1987), pp. 217-231, in Oper. Theory Adv. Appl. 35, Birkhäuser, Basel, 1988. | MR | Zbl

[RS2] R. Rochberg - S. Semmes, Nearly weakly orthonormal sequences, singular value estimates, and Calderón-Zygmund operators, J. Funct. Anal. 86 (1989), 237-306. | MR | Zbl

[Se] S. Semmes, Quasisymmetry, measure, and a question of Heinonen, Rev. Mat. Iberoamericana 12 (1996), 727-781. | MR | Zbl

[Sh] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), 243-279. | MR | Zbl

[St] E. Stein, “Singular Integrals and Differentiability Properties of Functions”, Princeton Univ. Press, Princeton, N.J., 1970. | MR | Zbl

[SW] E. Stein - G. Weiss, “Introduction to Fourier Analysis on Euclidean Spaces”, Princeton Univ. Press, Princeton, N.J., 1971. | MR | Zbl

[Su] D. Sullivan, Hyperbolic geometry and homeomorphisms, In: “Geometric Topology” (Proc. Georgia Topology Conf. Athens, Ga, 1977), pp. 543-555, Academic Press, New York, 1979. | MR | Zbl

[TV] P. Tukia J. Väisälä, Lipschitz and quasiconformal approximation and extension, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 303-342. | MR | Zbl

[V1] J. Väisälä, “Lectures on $n$ -dimensional Quasiconformal Mappings”, Lecture Notes in Math. 229, Springer-Verlag, Berlin, 1971. | MR | Zbl

[V2] J. Väisälä, Quasi-symmetric embeddings in Euclidean spaces, Trans. Amer. Math. Soc. 264 (1981), 191-204. | MR | Zbl

[V3] J. Väisälä, The wall conjecture on domains in Euclidean spaces, Manuscripta Math. 93 (1997), 515-534. | MR | Zbl

[Vu] M. Vuorinen, “Conformal Geometry and Quasiregular Mappings”, Lecture Notes in Math. 1319, Springer-Verlag, Berlin, 1988. | MR | Zbl

[W] P. Wojtaszczyk, “Banach Spaces for Analysts”, Cambridge Studies in adv. math. 25, Cambridge University Press, 1991. | MR | Zbl

[Y] K. Yosida, “Functional Analysis”, Springer-Verlag, Berlin-Heidelberg-New York, 1965. | Zbl

[Zie] W. P. Ziemer, Extremal length and $p$ -capacity, Mich. Math. J. 16 (1969), 43-51. | MR | Zbl

[Zin] M. Zinsmeister, A distortion theorem for quasiconformal mappings, Bull. Soc. Math. France 114 (1986), 123-133. | Numdam | MR | Zbl

[Zo] V. A. Zorich, Angular boundary values of quasiconformal mappings of a ball, Dokl. Akad. Nauk. SSSR 179 (1967), 1492-1494. | MR | Zbl