Quasiconformal maps in metric spaces with controlled geometry

Assouad, P., Plongements lipschitziens dansR n .Bull. Soc. Math. France, 111 (1983), 429–448.

Beurling, A. &Ahlfors, L. V., The boundary correspondence under quasiconformal mappings,Acta Math., 96 (1956), 125–142.

Bojarski, B., Homeomorphic solutions of Beltrami systems.Dokl. Akad. Nauk SSSR, 102 (1955), 661–664 (Russian).

Bourdon, M. & Pajot, H., Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings. To appear inProc. Amer. Math. Soc.

Buser, P., A note on the isoperimetric constant.Ann. Sci. École Norm. Sup. (4), 15 (1982), 213–230.

Cannon, J. W., The combinatorial Riemann mapping theorem.Acta Math., 173 (1994), 155–234.

Cheeger, J. &Colding, T. H., Almost rigidity of warped products and the structure of spaces with Ricci curvature bounded below.C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 353–357.

Coifman, R. &Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals.Studia Math., 51 (1974), 241–250.

Cannon, J. W., Floyd, W. J. &Parry, W. R., Squaring rectangles: The finite Riemann mapping theorem.Contemp. Math., 169 (1994), 133–212.

Chavel, I.,Riemannian Geometry—A Modern Introduction. Cambridge Tracts in Math., 108. Cambridge Univ. Press, Cambridge, 1993.

Cannon, J. W. &Swenson, E. L., Recognizing constant curvature discrete groups in dimension 3.Trans. Amer. Math. Soc., 350 (1998), 809–849.

Coulhon, T. &Saloff-Coste, L., Variétés riemanniennes isométriques à l'infini.Rev. Mat. Iberoamericana, 11 (1995), 687–726.

Coifman, R. &Weiss, G.,Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Math., 242. Springer-Verlag, Berlin-New York, 1971.

David, G. &Semmes, S., StrongA ∞-weights, Sobolev inequalities and quasiconformal mappings, inAnalysis and Partial Differential Equations, pp. 101–111. Lecture Notes in Pure and Appl. Math., 122. Dekker, New York, 1990.

—,Singular Integrals and Rectifiable Sets in R n :Au-delà des graphes lipschitziens. Astérisque, 193. Soc. Math. France, Paris, 1991.

—,Analysis of and on Uniformly Rectifiable Sets. Math. Surveys Monogr., 38. Amer. Math. Soc., Providence, RI, 1993.

Federer, H.,Geometric Measure Theory. Grundlehren Math. Wiss., 153. Springer-Verlag, New York, 1969.

Fuglede, B., Extremal length and functional completion.Acta Math., 98 (1957), 171–219.

Gehring, F. W., The definitions and exceptional sets for quasiconformal mappings.Ann. Acad. Sci. Fenn. Math., 281 (1960), 1–28.

—, Symmetrization of rings in space.Trans. Amer. Math. Soc., 101 (1961), 499–519.

—, TheL P-integrability of the partial derivatives of a quasiconformal mapping.Acta Math., 130 (1973), 265–277.

Ghys, E. &Harpe, P. de la,Sur les groupes hyperboliques d'après Mikhael Gromov. Progr. Math., 83. Birkhäuser, Boston, MA, 1990.

Gromov, M.,Structures métriques pour les variétés riemanniennes. Edited by J. Lafontaine and P. Pansu. Textes Mathématiques, 1. CEDIC, Paris, 1981.

Gehring, F. W. &Martio, O., Quasiextremal distance domains and extension of quasiconformal mappings.J. Analyse Math., 45 (1985), 181–206.

Gromov, M. &Pansu, P., Rigidity of lattices: An introduction, inGeometric Topology: Recent Developments. Lecture Notes in Math., 1504. Springer-Verlag, Berlin-New York, 1991.

Gromov, M., Carnot-Carathéodory spaces seen from within, inSub-Riemannian Geometry, pp. 79–323. Progr. Math., 144. Birkhäuser, Basel, 1996.

Gilbarg, D. &Trudinger, N. S.,Elliptic Partial Differential Equations of Second Order, 2nd edition. Grundlehren Math. Wiss., 224. Springer-Verlag, Berlin-New York, 1983.

Heinonen, J., Quasiconformal mappings onto John domains.Rev. Mat. Iberoamericana, 5 (1989), 97–123.

—, A capacity estimate on Carnot groups.Bull. Sci. Math., 119 (1995), 475–484.

Haj⦊asz, P., Sobolev spaces on an arbitrary metric space.Potential Anal., 5 (1996), 403–415.

Haj⦊asz, P. &Koskela, P., Sobolev meets Poincaré.C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1211–1215.

Hesse, J., Ap-extremal length andp-capacity equality.Ark. Mat., 13 (1975), 131–144.

Heinonen, J. &Koskela, P., Definitions of quasiconformality.Invent. Math., 120 (1995), 61–79.

—, From local to global in quasiconformal structures.Proc. Nat. Acad. Sci. U.S.A., 93 (1996), 554–556.

Heinonen, J., Kilpeläinen, T. &Martio, O.,Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Univ. Press, New York, 1993.

Hocking, J. G. &Young, G. S.,Topology. Academic Press, Reading, MA, 1959.

Jerison, D., The Poincaré inequality for vector fields satisfying Hörmander's condition.Duke Math. J., 53 (1986), 503–523.

Koskela, P., Removable sets for Sobolev spaces. To appear inArk. Mat.

Koskela, P. &MacManus, P., Quasiconformal mappings and Sobolev spaces.Studia Math., 131 (1998), 1–17.

Korányi, A. &Reimann, H. M., Quasiconformal mappings on the Heisenberg group.Invent. Math., 80 (1985), 309–338.

—, Foundations for the theory of quasiconformal mappings on the Heisenberg group.Adv. in Math., 111 (1995), 1–87.

Korevaar, J. N. &Schoen, R. M., Sobolev spaces and harmonic maps for metric space targets.Comm. Anal. Geom., 1 (1993), 561–659.

Loewner, C., On the conformal capacity in space.J. Math. Mech., 8 (1959), 411–414.

Mattila, P.,Geometry of Sets and Measures in Euclidean Spaces. Cambridge Stud. Adv. Math., 44. Cambridge Univ. Press, Cambridge, 1995.

Margulis, G. A. &Mostow, G. D., The differential of a quasiconformal mapping of a Carnot-Carathéodory space.,Geom. Funct. Anal., 5 (1995), 402–433.

Mostow, G. D.,Strong Rigidity of Locally Symmetric Spaces. Princeton Univ. Press, Princeton, NJ, 1973.

—, A remark on quasiconformal mappings on Carnot groups.Michigan Math. J., 41 (1994), 31–37.

Maheux, P. &Saloff-Coste, L., Analyse sur les boules d'un opérateur souselliptique.Math. Ann., 303 (1995), 713–740.

Näkki, R., Extension of Loewner's capacity theorem.Trans. Amer. Math. Soc., 180 (1973), 229–236.

Pansu, P., Quasiisométries des variétés à courbure négative. Thesis, Université Paris VII, 1987.

—, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un.Ann. of Math., 129 (1989), 1–60.

—, Dimension conforme et sphère à l'infini des variétés à courbure négative.Ann. Acad. Sci. Fenn. Math., 14 (1989), 177–212.

Paulin, P., Un groupe hyperbolique est déterminé par son bord.J. London Math. Soc., 54 (1996), 50–74.

Reimann, H. M., An estimate for pseudoconformal capacities on the sphere.Ann. Acad. Sci. Fenn. Math., 14 (1989), 315–324.

Rudin, W.,Real and Complex Analysis, 2nd edition. McGraw-Hill, New York-Düsseldorf-Johannesburg, 1974.

Semmes, S., Bilipschitz mappings and strongA ∞-weights.Ann. Acad. Sci. Fenn. Math., 18 (1993), 211–248.

—, On the nonexistence of bilipschitz parameterizations and geometric problems aboutA ∞-weights.Rev. Mat. Iberoamericana, 12 (1996), 337–410.

—, Good metric spaces without good parameterizations.Rev. Math. Iberoamericana, 12 (1996), 187–275.

—, Finding curves on general spaces through quantitative topology with applications to Sobolev and Poincarè inequalities.Selecta Math. (N.S.), 2 (1996), 155–295.

—, Some remarks about metric spaces, spherical mappings, functions and their derivatives.Publ. Math., 40 (1996), 411–430.

Saloff-Coste, L., Uniformly elliptic operators on Riemannian manifolds.J. Differential Geom., 36 (1992), 417–450.

Stein, E. M.,Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ, 1970.

—,Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton Univ. Press, Princeton, NJ, 1993.

Strömberg, J. D. &Torchinsky, A.,Weighted Hardy Spaces, Lecture Notes in Math., 1381. Springer-Verlag, Berlin-New York, 1989.

Sullivan, D. P., Hyperbolic geometry and homeomorphisms, inGeometric Topology (Athens, GA, 1977), pp. 543–555. Academic Press, New York-London, 1979.

Tukia, P. &Väisälä, J., Quasisymmetric embeddings of metric spaces.Ann. Acad. Sci. Fenn. Math., 5 (1980), 97–114.

Tyson, J., Quasiconformality and quasisymmetry in metric measure spaces. To appear inAnn. Acad. Sci. Fenn. Math.

Väisälä, J.,Lectures on n-Dimensional Quasiconformal Mappings. Lecture Notes in Math., 229. Springer-Verlag, Berlin-New York, 1971.

—, Quasisymmetric embeddings in Euclidean spaces.Trans. Amer. Math. Soc., 264 (1981), 191–204.

—, Quasimöbius maps.J. Analyse Math., 44 (1984/85), 218–234.

—, Quasiconformal maps of cylindrical domains.Acta Math., 162 (1989), 201–225.

—, Free quasiconformality in Banach spaces, I.Ann. Acad. Sci. Fenn. Math., 15 (1990), 355–379.

—, Free quasiconformality in Banach spaces, II.Ann. Acad. Sci. Fenn. Math., 16 (1991), 255–310.

Vodop'yanov, S. K. &Greshnov, A. V., Analytic properties of quasiconformal mappings on Carnot groups.Siberian Math. J., 36 (1995), 1142–1151.

Varopoulos, N. Th., Saloff-Coste, L. &Coulhon, T.,Analysis and Geometry on Groups. Cambridge Tracts in Math., 100. Cambridge Univ. Press, Cambridge, 1992.

Vuorinen, M.,Conformal Geometry and Quasiregular Mappings. Lecture Notes in Math., 1319. Springer-Verlag, New York, 1988.

Ziemer, W. P., Extremal length andp-capacity.Michigan Math. J., 16 (1969), 43–51.