Differentiating maps into \(L^1\), and the geometry of BV functions. (English) Zbl 1194.22009

Cheeger, Jeff; Kleiner, Bruce

Ann. Math. (2) 171, No. 2, 1347-1385 (2010).

Considering a metric measure space \(X\) and a Banach space \(V\) the authors examine the connection between differentiation theory for Lipschitz maps \(X\to V\) and bi-Lipschitz non-embeddability. The paper is devoted to the case when \(V=L^1\), where the differentiability fails. Supposing that the space \(X\) carries a measure, the authors give some alternative characterizations of \(L^1\) maps \(f\to L^1\) and discuss the equivalence between metrics induced by maps to \(L^1\) and cut metrics. The total perimeter measure \(\lambda\in \text{Radon}(X)\) associated to a finite perimeter cut measure is constructed and its properties are studied taking into account location and scale.
The notion of bad part of \(\lambda\) is also defined and it is the key in proving the main differentiation theorem given in this paper. A new type of differentiability is established for certain \(X\), including \(R^n\) and the Heisenberg group \(H\) with its Carnot-Carathéodory metric. It is shown that \(H\) does not bi-Lipschitz embed into \(L^1\). This result provides a natural counterexample to Goemans-Linial conjecture in theoretical computer science. The new connection established between the Lipschitz maps to \(L^1\) and the functions of bounded variation (BV) allows to exploit different results on the structure of BV functions on the Heisenberg group.

Reviewer: Gheorghe Zet (Iaşi)

MSC:

22E30	Analysis on real and complex Lie groups
26B30	Absolutely continuous real functions of several variables, functions of bounded variation
46B20	Geometry and structure of normed linear spaces
49Q15	Geometric measure and integration theory, integral and normal currents in optimization

Keywords:

Lipschitz map; differentiation theory; total perimeter; Heisenberg group; function of bounded variation

References:

[1]	L. Ambrosio, ”Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces,” Adv. Math., vol. 159, iss. 1, pp. 51-67, 2001. · Zbl 1002.28004
[2]	L. Ambrosio, ”Fine properties of sets of finite perimeter in doubling metric measure spaces,” Set-Valued Anal., vol. 10, iss. 2-3, pp. 111-128, 2002. · Zbl 1037.28002
[3]	L. Ambrosio, M. Miranda Jr., and D. Pallara, ”Special functions of bounded variation in doubling metric measure spaces,” in Calculus of Variations: Topics From the Mathematical Heritage of E. De Giorgi, Dept. Math., Seconda Univ. Napoli, Caserta, 2004, pp. 1-45. · Zbl 1089.49039
[4]	N. Aronszajn, ”Differentiability of Lipschitzian mappings between Banach spaces,” Studia Math., vol. 57, iss. 2, pp. 147-190, 1976. · Zbl 0342.46034
[5]	Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, Providence, RI: Amer. Math. Soc., 2000. · Zbl 0946.46002
[6]	M. Bourdon and H. Pajot, ”Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings,” Proc. Amer. Math. Soc., vol. 127, iss. 8, pp. 2315-2324, 1999. · Zbl 0924.30030
[7]	J. Cheeger, ”Differentiability of Lipschitz functions on metric measure spaces,” Geom. Funct. Anal., vol. 9, iss. 3, pp. 428-517, 1999. · Zbl 0942.58018
[8]	J. Cheeger and B. Kleiner, ”On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces,” in Inspired by S. S. Chern, World Sci. Publ., Hackensack, NJ, 2006, pp. 129-152. · Zbl 1139.58004
[9]	J. Cheeger and B. Kleiner, ”Generalized differential and bi-Lipschitz nonembedding in \(L^1\),” C. R. Math. Acad. Sci. Paris, vol. 343, iss. 5, pp. 297-301, 2006. · Zbl 1100.58004
[10]	J. Cheeger and B. Kleiner, ”On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces,” in Inspired by S. S. Chern, World Sci. Publ., Hackensack, NJ, 2006, pp. 129-152. · Zbl 1139.58004
[11]	J. Cheeger and B. Kleiner, ”Characterization of the Radon-Nikodym Property in terms of inverse limits,” in Géométrie Différentielle, Physique Mathématique, Mathématiques et Société pour Célébrer les 60 ans de Jean-Pierre Bourguignon, Séminaires et Congrès, Société Mathématique de France, 2008. · Zbl 1191.46016
[12]	Cheeger, Jeff and Kleiner, Bruce, Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodym Property, 2008. · Zbl 1200.58007
[13]	Cheeger, Jeff and Kleiner, Bruce, Metric differentiation, monotonicity and Lipschitz maps to \(L^1\), 2008. · Zbl 1214.46013
[14]	Cheeger, J. and Kleiner, B. and A. Naor, Quantitative bounds on the rate of central collapse for BV maps from the Heisenberg group to \(L^1\). · Zbl 1247.46020
[15]	E. De Giorgi, ”Nuovi teoremi relativi alle misure \((r-1)\)-dimensionali in uno spazio ad \(r\) dimensioni,” Ricerche Mat., vol. 4, pp. 95-113, 1955. · Zbl 0066.29903
[16]	M. M. Deza and M. Laurent, Geometry of Cuts and Metrics, New York: Springer-Verlag, 1997. · Zbl 0885.52001
[17]	B. Franchi, R. Serapioni, and F. Serra Cassano, ”Rectifiability and perimeter in the Heisenberg group,” Math. Ann., vol. 321, iss. 3, pp. 479-531, 2001. · Zbl 1057.49032
[18]	B. Franchi, R. Serapioni, and F. Serra Cassano, ”On the structure of finite perimeter sets in step 2 Carnot groups,” J. Geom. Anal., vol. 13, iss. 3, pp. 421-466, 2003. · Zbl 1064.49033
[19]	E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Boston: Birkhäuser, 1984. · Zbl 0545.49018
[20]	M. Gromov, ”Carnot-Carathéodory spaces seen from within,” in Sub-Riemannian Geometry, Boston: Birkhäuser, 1996, pp. 79-323. · Zbl 0864.53025
[21]	J. Heinonen and P. Koskela, ”From local to global in quasiconformal structures,” Proc. Nat. Acad. Sci. U.S.A., vol. 93, iss. 2, pp. 554-556, 1996. · Zbl 0842.30016
[22]	J. Heinonen, P. Koskela, N. Shanmugalingam, and J. T. Tyson, ”Sobolev classes of Banach space-valued functions and quasiconformal mappings,” J. Anal. Math., vol. 85, pp. 87-139, 2001. · Zbl 1013.46023
[23]	S. Heinrich and P. Mankiewicz, ”Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces,” Studia Math., vol. 73, iss. 3, pp. 225-251, 1982. · Zbl 0506.46008
[24]	S. Kakutani, ”Mean ergodic theorem in abstract \((L)\)-spaces,” Proc. Imp. Acad., Tokyo, vol. 15, pp. 121-123, 1939. · Zbl 0021.41304
[25]	B. Kirchheim, ”Rectifiable metric spaces: local structure and regularity of the Hausdorff measure,” Proc. Amer. Math. Soc., vol. 121, iss. 1, pp. 113-123, 1994. · Zbl 0806.28004
[26]	S. Khot and N. Vishnoi, ”The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into \(\ell_1\),” in 46th Annual Symposium on Foundations of Computer Science, Los Alamitos: IEE Computer Society, 2005, pp. 53-62. · Zbl 1321.68316
[27]	T. J. Laakso, ”Ahlfors \(Q\)-regular spaces with arbitrary \(Q>1\) admitting weak Poincaré inequality,” Geom. Funct. Anal., vol. 10, iss. 1, pp. 111-123, 2000. · Zbl 0962.30006
[28]	Lee, J. R. and Naor, A., \(L^p\) metrics on the Heisenberg group and the Goemans-Linial conjecture, 2006.
[29]	J. Lindenstrauss and D. Preiss, ”Fréchet differentiability of Lipschitz functions (a survey),” in Recent Progress in Functional Analysis (Valencia, 2000), Amsterdam: North-Holland, 2001, pp. 19-42. · Zbl 1037.46043
[30]	M. Miranda Jr., ”Functions of bounded variation on “good” metric spaces,” J. Math. Pures Appl., vol. 82, iss. 8, pp. 975-1004, 2003. · Zbl 1109.46030
[31]	P. Pansu, ”Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un,” Ann. of Math., vol. 129, iss. 1, pp. 1-60, 1989. · Zbl 0678.53042
[32]	S. D. Pauls, ”The large scale geometry of nilpotent Lie groups,” Comm. Anal. Geom., vol. 9, iss. 5, pp. 951-982, 2001. · Zbl 1005.53033
[33]	S. Semmes, ”On the nonexistence of bi-Lipschitz parameterizations and geometric problems about \(A_\infty\)-weights,” Rev. Mat. Iberoamericana, vol. 12, iss. 2, pp. 337-410, 1996. · Zbl 0858.46017
[34]	W. P. Ziemer, Weakly Differentiable Functions, New York: Springer-Verlag, 1989. · Zbl 0692.46022