Abstract
The paper presents a study of Fuglede’s \(p\)-module of systems of measures in condensers in polarizable Carnot groups. In particular, we calculate the \(p\)-module of measures in spherical ring domains, find the extremal measures, and finally, extend a theorem by Rodin to these groups.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aikawa, H. and Ohtsuka, M., Extremal length of vector measures, Ann. Acad. Sci. Fenn. Math. 24 (1999), 61–88.
Balogh, Z. M. and Tyson, J. T., Polar coordinates in Carnot groups, Math. Z. 241 (2002), 697–730.
Brakalova, M. A., Boundary extension of \(\mu \)–homeomorphisms, in Complex and Harmonic Analysis, pp. 231–247, DEStech, Lancaster, PA, 2007.
Brakalova, M., Markina, I. and Vasil’ev, A., Extremal functions for modules of systems of measures, arXiv:1409.1626 [math.CA], 2014.
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.
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.
Chang, D. C. and Markina, I., Anisotropic quaternion Carnot groups: geometric analysis and Green function, Adv. in Appl. Math. 39 (2007), 345–394.
Federer, H., Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften 153, Springer, New York, 1969.
Folland, G. B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161–207.
Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Groups, Mathematical Notes 28, Princeton Univ. Press, Princeton, NJ, 1982.
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.
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. Spaces 2 (2014), 258–281.
Fuglede, B., Extremal length and functional completion, Acta Math. 98 (1957), 171–219.
Gehring, F. W., Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961), 499–519.
Gehring, F. W., Extremal length definitions for the conformal capacity of rings in space, Michigan Math. J. 9 (1962), 137–150.
Gutlyanskii, V.Ya., Martio, O., Sugawa, T. and Vuorinen, M., On the degenerate Beltrami equation, Trans. Amer. Math. Soc. 357 (2005), 875–900.
Gutlyanskii, V.Ya., Sakan, K. and Sugawa, T., On \(\mu \)-conformal homeomorphisms and boundary correspondence, Complex Var. Elliptic Equ. 58 (2013), 947–962.
Heinonen, J., Calculus on Carnot groups, in Fall School in Analysis, Juväskylä, 1994, Ber. Univ. Juväskylä Math. Inst. 68, pp. 1–31, 1995.
Heinonen, J. and Holopainen, I., Quasiregular maps on Carnot groups, J. Geom. Anal. 7 (1997), 109–148.
Hesse, J., A \(p\)-extremal length and \(p\)-capacity equality, Ark. Mat. 13 (1975), 131–144.
Kaplan, A., Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147–153.
Karmanova, M. and Vodop’yanov, S., Carnot-Carathéodory spaces, coarea and area formulas, in Analysis and Mathematical Physics, Trends in Mathematics, pp. 233–335, Birkhäuser, Basel, 2009.
Korányi, A. and Reimann, H. M., Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), 309–338.
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.
Krivov, V. V., Some properties of moduli in space, Dokl. Akad. Nauk SSSR 154 (1964), 510–513.
Loewner, Ch., On the conformal capacity in space, J. Math. Mech. 8 (1959), 411–414.
Magnani, V., The coarea formula for real-valued Lipschitz maps on stratified groups, Math. Nachr. 278 (2005), 1689–1705.
Markina, I., On coincidence of \(p\)-module of a family of curves and \(p\)-capacity on the Carnot group, Rev. Mat. Iberoam. 19 (2003), 143–160.
Markina, I., Extremal widths on homogeneous groups, Complex Var. Theory Appl. 48 (2003), 947–960.
Markina, I., \(p\)-module of vector measures in domains with intrinsic metric on Carnot groups, Tohoku Math. J. (2) 56 (2004), 553–569.
Markina, I., Modules of vector measures on the Heisenberg group, in Complex Analysis and Dynamical Systems II, Contemp. Math. 382, pp. 291–304, Amer. Math. Soc., Providence, RI, 2005.
Mitchell, J., On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), 35–45.
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.
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
Pansu, P., Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), 1–60 (in French).
Platis, I. D., Modulus of surface families and the radial stretch in the Heisenberg group, arXiv:1310.4292.
Platis, I. D., Modulus of revolution rings in the Heisenberg group, arXiv:1504.05099.
Rodin, B., The method of extremal length, Bull. Amer. Math. Soc. 80 (1974), 587–606.
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.
Rodin, B. and Warschawski, S. E., Extremal length and univalent functions. I. The angular derivative, Math. Z. 153 (1977), 1–17.
Šabat, B. V., The modulus method in space, Dokl. Akad. Nauk SSSR 130 (1960), 1210–1213. (in Russian).
Shanmugalingam, N., Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), 243–279.
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).
Shlyk, V. A., On the equality between \(p\)-capacity and \(p\)-modulus, Sibirsk. Mat. Zh. 34 (1993), 216–221 (in Russian).
Väisälä, J., On quasiconformal mapping in space, Ann. Acad. Sci. Fenn. Ser. AI 296 (1961), 1–36.
Vodopyanov, S. K., Potential theory on homogeneous groups, Mat. Sb. 180 (1989), 57–77.
Ziemer, W. P., Extremal length and conformal capacity, Trans. Amer. Math. Soc. 126 (1967), 460–473.
Ziemer, W. P., Extremal length and \(p\)-capacity, Michigan Math. J. 16 (1969), 43–51.
Ziemer, W. P., Extremal length as a capacity, Michigan Math. J. 17 (1970), 117–128.
Author information
Authors and Affiliations
Corresponding author
Additional information
M. Brakalova was partially supported by a Faculty Research Grant, Fordham University, USA. I. Markina and A. Vasil’ev were partially supported by the grants of the Norwegian Research Council #239033/F20 and #213440/BG, as well as by EU FP7 IRSES program STREVCOMS, grant no. PIRSES-GA-2013-612669.
Rights and permissions
About this article
Cite this article
Brakalova, M., Markina, I. & Vasil’ev, A. Modules of systems of measures on polarizable Carnot groups. Ark Mat 54, 371–401 (2016). https://doi.org/10.1007/s11512-016-0242-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11512-016-0242-6