×

Modules of systems of measures on polarizable Carnot groups. (English) Zbl 1367.30042

Fuglede’s \(p\)-module of systems of measures is studied in the context of polarizable Carnot groups. Spherical ring domains are studied in detail and for instance a formula is given for the \(p\)-module of the family of curves joing the boundary components of such a ring.

MSC:

30L10 Quasiconformal mappings in metric spaces
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
Full Text: DOI

References:

[1] Aikawa, H. and Ohtsuka, M., Extremal length of vector measures, Ann. Acad. Sci. Fenn. Math.24 (1999), 61-88. · Zbl 0940.31006
[2] Balogh, Z. M. and Tyson, J. T., Polar coordinates in Carnot groups, Math. Z.241 (2002), 697-730. · Zbl 1015.22005 · doi:10.1007/s00209-002-0441-7
[3] Brakalova, M. A., Boundary extension of μ \(\mu \)-homeomorphisms, 231-247 (2007), Lancaster, PA · Zbl 1151.30018
[4] Brakalova, M., Markina, I. and Vasil’ev, A., Extremal functions for modules of systems of measures, arXiv:1409.1626 [math.CA], 2014. · Zbl 1387.30026
[5] Calin, O., Chang, D. C. and Greiner, P., Geometric Analysis on the Heisenberg Group and Its Generalizations, AMS/IP Studies in Advanced Mathematics 40, Amer. Math. Soc./International Press, Providence, RI/Somerville, MA, 2007. · Zbl 1132.53001
[6] Capogna, L., Danielli, D. and Garofalo, N., Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math.118 (1996), 1153-1196. · Zbl 0878.35020 · doi:10.1353/ajm.1996.0046
[7] Chang, D. C. and Markina, I., Anisotropic quaternion Carnot groups: geometric analysis and Green function, Adv. in Appl. Math.39 (2007), 345-394. · Zbl 1133.35334 · doi:10.1016/j.aam.2007.02.002
[8] Federer, H., Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften 153, Springer, New York, 1969. · Zbl 0176.00801
[9] Folland, G. B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat.13 (1975), 161-207. · Zbl 0312.35026 · doi:10.1007/BF02386204
[10] Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Groups, Mathematical Notes 28, Princeton Univ. Press, Princeton, NJ, 1982. · Zbl 0508.42025
[11] Franchi, B., Serapioni, R. and Serra Cassano, F., Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups, Comm. Anal. Geom.11 (2003), 909-944. · Zbl 1077.22008 · doi:10.4310/CAG.2003.v11.n5.a4
[12] Franchi, B., Marchi, M., Serapioni, R. P. and Serra Cassano, F., Differentiability and approximate differentiability for intrinsic Lipschitz functions in Carnot groups and a Rademacher theorem, Anal. Geom. Metr. Spaces2 (2014), 258-281. · Zbl 1307.22007
[13] Fuglede, B., Extremal length and functional completion, Acta Math.98 (1957), 171-219. · Zbl 0079.27703 · doi:10.1007/BF02404474
[14] Gehring, F. W., Symmetrization of rings in space, Trans. Amer. Math. Soc.101 (1961), 499-519. · Zbl 0104.30002 · doi:10.1090/S0002-9947-1961-0132841-2
[15] Gehring, F. W., Extremal length definitions for the conformal capacity of rings in space, Michigan Math. J.9 (1962), 137-150. · Zbl 0109.04904 · doi:10.1307/mmj/1028998672
[16] Gutlyanskii, V.Ya., Martio, O., Sugawa, T. and Vuorinen, M., On the degenerate Beltrami equation, Trans. Amer. Math. Soc.357 (2005), 875-900. · Zbl 1066.30021 · doi:10.1090/S0002-9947-04-03708-0
[17] Gutlyanskii, V.Ya., Sakan, K. and Sugawa, T., On μ \(\mu \)-conformal homeomorphisms and boundary correspondence, Complex Var. Elliptic Equ.58 (2013), 947-962. · Zbl 1291.30126 · doi:10.1080/17476933.2011.613116
[18] Heinonen, J., Calculus on Carnot groups, Juväskylä, 1994 · Zbl 0863.22009
[19] Heinonen, J. and Holopainen, I., Quasiregular maps on Carnot groups, J. Geom. Anal.7 (1997), 109-148. · Zbl 0905.30018 · doi:10.1007/BF02921707
[20] Hesse, J., A p \(p\)-extremal length and p \(p\)-capacity equality, Ark. Mat.13 (1975), 131-144. · Zbl 0302.31009 · doi:10.1007/BF02386202
[21] Kaplan, A., Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc.258 (1980), 147-153. · Zbl 0393.35015 · doi:10.1090/S0002-9947-1980-0554324-X
[22] Karmanova, M.; Vodop’yanov, S., Carnot-Carathéodory spaces, coarea and area formulas, 233-335 (2009), Basel · Zbl 1297.53029 · doi:10.1007/978-3-7643-9906-1_14
[23] Korányi, A. and Reimann, H. M., Quasiconformal mappings on the Heisenberg group, Invent. Math.80 (1985), 309-338. · Zbl 0567.30017 · doi:10.1007/BF01388609
[24] Korányi, A. and Reimann, H. M., Horizontal normal vectors and conformal capacity of spherical rings in the Heisenberg group, Bull. Sci. Math. (2)111 (1987), 3-21. · Zbl 0629.22006
[25] Krivov, V. V., Some properties of moduli in space, Dokl. Akad. Nauk SSSR154 (1964), 510-513. · Zbl 0166.33401
[26] Loewner, Ch., On the conformal capacity in space, J. Math. Mech.8 (1959), 411-414. · Zbl 0086.28203
[27] Magnani, V., The coarea formula for real-valued Lipschitz maps on stratified groups, Math. Nachr.278 (2005), 1689-1705. · Zbl 1079.49030 · doi:10.1002/mana.200310334
[28] Markina, I., On coincidence of p \(p\)-module of a family of curves and p \(p\)-capacity on the Carnot group, Rev. Mat. Iberoam.19 (2003), 143-160. · Zbl 1038.31006 · doi:10.4171/RMI/340
[29] Markina, I., Extremal widths on homogeneous groups, Complex Var. Theory Appl.48 (2003), 947-960. · Zbl 1048.31004
[30] Markina, I., p \(p\)-module of vector measures in domains with intrinsic metric on Carnot groups, Tohoku Math. J. (2)56 (2004), 553-569. · Zbl 1069.31003 · doi:10.2748/tmj/1113246750
[31] Markina, I., Modules of vector measures on the Heisenberg group, No. 382, 291-304 (2005), Providence, RI · Zbl 1087.31007 · doi:10.1090/conm/382/07069
[32] Mitchell, J., On Carnot-Carathéodory metrics, J. Differential Geom.21 (1985), 35-45. · Zbl 0554.53023
[33] Montefalcone, F., Sets of finite perimeter associated with vector fields and polyhedral approximation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.14 (2003), 279-295. 2004. · Zbl 1072.49031
[34] Ohtsuka, M., Extremal Length and Precise Functions, GAKUTO International Series. Mathematical Sciences and Applications 19, Gakkōtosho, Tokyo, 2003, with a preface by Fumi-Yuki Maeda · Zbl 1075.31001
[35] Pansu, P., Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2)129 (1989), 1-60 (in French). · Zbl 0678.53042 · doi:10.2307/1971484
[36] Platis, I. D., Modulus of surface families and the radial stretch in the Heisenberg group, arXiv:1310.4292. · Zbl 1375.30027
[37] Platis, I. D., Modulus of revolution rings in the Heisenberg group, arXiv:1504.05099. · Zbl 1351.30043
[38] Rodin, B., The method of extremal length, Bull. Amer. Math. Soc.80 (1974), 587-606. · Zbl 0286.30014 · doi:10.1090/S0002-9904-1974-13517-2
[39] Rodin, B. and Warschawski, S. E., Extremal length and the boundary behavior of conformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math.2 (1976), 467-500. · Zbl 0348.30007 · doi:10.5186/aasfm.1976.0231
[40] Rodin, B. and Warschawski, S. E., Extremal length and univalent functions. I. The angular derivative, Math. Z.153 (1977), 1-17. · Zbl 0384.30006 · doi:10.1007/BF01214728
[41] Šabat, B. V., The modulus method in space, Dokl. Akad. Nauk SSSR130 (1960), 1210-1213. (in Russian). · Zbl 0096.19604
[42] Shanmugalingam, N., Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam.16 (2000), 243-279. · Zbl 0974.46038 · doi:10.4171/RMI/275
[43] Shlyk, V. A., Capacity of a condenser and the modulus of a family of separating surfaces, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)185 (1990), 168-182 (in Russian). · Zbl 0734.31008
[44] Shlyk, V. A., On the equality between p \(p\)-capacity and p \(p\)-modulus, Sibirsk. Mat. Zh.34 (1993), 216-221 (in Russian). · Zbl 0810.31004
[45] Väisälä, J., On quasiconformal mapping in space, Ann. Acad. Sci. Fenn. Ser. AI296 (1961), 1-36. · Zbl 0096.27506
[46] Vodopyanov, S. K., Potential theory on homogeneous groups, Mat. Sb.180 (1989), 57-77. · Zbl 0673.31005
[47] Ziemer, W. P., Extremal length and conformal capacity, Trans. Amer. Math. Soc.126 (1967), 460-473. · Zbl 0177.34002 · doi:10.1090/S0002-9947-1967-0210891-0
[48] Ziemer, W. P., Extremal length and p \(p\)-capacity, Michigan Math. J.16 (1969), 43-51. · Zbl 0172.38701 · doi:10.1307/mmj/1029000164
[49] Ziemer, W. P., Extremal length as a capacity, Michigan Math. J.17 (1970), 117-128. · Zbl 0183.39104 · doi:10.1307/mmj/1029000421
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.