Differentiability of Intrinsic Lipschitz Functions within Heisenberg Groups

Bruno Franchi1, Raul Serapioni2, Francesco Serra Cassano2
1Dipartimento di Matematica, Università di Bologna, Bologna, Italy
2Dipartimento di Matematica, Università di Trento, Povo (Trento), Italy

Tóm tắt

Từ khóa


Tài liệu tham khảo

Ambrosio, L., Kirchheim, B.: Rectifiable sets in metric and Banach spaces. Math. Ann. 318, 527–555 (2000)

Ambrosio, L., Kirchheim, B.: Currents in metric spaces. Acta Math. 185, 1–80 (2000)

Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, London (2000)

Arena, G., Serapioni, R.: Intrinsic regular submanifolds in Heisenberg groups are differentiable graphs. Calc. Var. Partial. Differ. Equ. 35(4), 517–536 (2009)

Bigolin, F., Vittone, D.: Some remarks about parametrizations of intrinsic regular surfaces in the Heisenberg group. Publ. Mat. 54(1), 159–172 (2010)

Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin (2007)

Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428–517 (1999)

Cheeger, J., Kleiner, B.: Generalized differential and bi-Lipschitz nonembedding in L 1. C. R. Math. Acad. Sci. Paris 343(5), 297–301 (2006)

Cheeger, J., Kleiner, B.: Differentiating maps into L 1 and the geometry of BV functions. Ann. Math. 171(2), 1347–1385 (2010)

Citti, G., Manfredini, M.: Blow-up in non-homogeneous Lie groups and rectifiability. Houst. J. Math. 31(2), 333–353 (2005)

Citti, G., Manfredini, M.: Implicit function theorem in Carnot–Carathéodory spaces. Commun. Contemp. Math. 8(5), 657–680 (2006)

Citti, G., Sarti, A.: A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vis. 24(3), 307–326 (2006)

Cole, D., Pauls, S.D.: C 1 hypersurfaces of the Heisenberg group are N-rectifiable. Houst. J. Math. 32(3), 307–326 (2006)

Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)

Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)

Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton (1982)

Franchi, B., Serapioni, R., Serra Cassano, F.: Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321, 479–531 (2001)

Franchi, B., Serapioni, R., Serra Cassano, F.: Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups. Commun. Anal. Geom. 11(5), 909–944 (2003)

Franchi, B., Serapioni, R., Serra Cassano, F.: On the structure of finite perimeter sets in step 2 Carnot groups. J. Geom. Anal. 13(3), 421–466 (2003)

Franchi, B., Serapioni, R., Serra Cassano, F.: Intrinsic Lipschitz graphs in Heisenberg groups. J. Nonlinear Convex Anal. 7(3), 423–441 (2006)

Franchi, B., Serapioni, R., Serra Cassano, F.: Regular submanifolds, graphs and area formula in Heisenberg groups. Adv. Math. 211(1), 152–203 (2007)

Garofalo, N., Nhieu, D.M.: Isoperimetric and Sobolev inequalities for Carnot–Carathéodory spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. 49, 1081–1144 (1996)

Gromov, M.: Carnot–Carathéodory spaces seen from within. In: Bellaïche, A., Risler, J. (eds.) Subriemannian Geometry. Progress in Mathematics, vol. 144. Birkhäuser, Basel (1996)

Heinonen, J.: Lectures on Lipschitz analysis. Report, University of Jyväskylä, Department of Mathematics and Statistics, 100. Jyväskylä (2005)

Heinonen, J.: Nonsmooth calculus. Bull. Am. Math. Soc. 44, 163–232 (2007)

Korányi, A., Reimann, H.M.: Foundation for the theory of quasiconformal mappings on the Heisenberg group. Adv. Math. 111, 1–87 (1995)

Lloyd, N.G.: Degree Theory. Cambridge Tracts in Mathematics, vol. 73. Cambridge University Press, Cambridge (1978)

Magnani, V.: Elements of Geometric Measure Theory on Sub-Riemannian Groups. Tesi di Perfezionamento. Scuola Normale Superiore, Pisa (2003)

Magnani, V.: Towards differential calculus in stratified groups. Preprint (2007)

Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)

Mattila, P., Serapioni, R., Serra Cassano, F.: Characterizations of intrinsic rectifiability in Heisenberg groups. Ann. Scuola Norm. Super. Pisa (to appear)

Mitchell, J.: On Carnot–Carathéodory metrics. J. Differ. Geom. 21, 35–45 (1985)

Pansu, P.: Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math. 129, 1–60 (1989)

Pauls, S.D.: A notion of rectifiability modeled on Carnot groups. Indiana Univ. Math. J. 53(1), 49–81 (2004)

Semmes, S.: On the non existence of biLipschitz parametrization and geometric problems about A ∞ weights. Rev. Mat. Iberoam. 12, 337–410 (1996)

Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton (1993)

Varopoulos, N.Th., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge University Press, Cambridge (1992)