Skip to main content Accessibility help
×
Home

POINCARÉ AND SOBOLEV INEQUALITIES FOR DIFFERENTIAL FORMS IN HEISENBERG GROUPS AND CONTACT MANIFOLDS

Abstract

In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups $\mathbb{H}^{n}$ , where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex $(E_{0}^{\bullet },d_{c})$ , which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of $\mathbb{H}^{n}$ . In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, which act trivially on cohomology. For instance, this allows us to replace a closed form, up to adding a controlled exact form, with a much more regular differential form.

Copyright

References

Hide All
1.Baldi, A., Barnabei, M. and Franchi, B., A recursive basis for primitive forms in symplectic spaces and applications to Heisenberg groups, Acta Math. Sin. (Engl. Ser.) 32(3) (2016), 265285. MR 3456421.
2.Baldi, A. and Franchi, B., Sharp a priori estimates for div-curl systems in Heisenberg groups, J. Funct. Anal. 265(10) (2013), 23882419. MR 3091819.
3.Baldi, A., Franchi, B. and Pansu, P., Gagliardo-Nirenberg inequalities for differential forms in Heisenberg groups, Math. Ann. 365(3–4) (2016), 16331667. MR 3521101.
4.Baldi, A., Franchi, B. and Pansu, P., L 1 -Poincaré and Sobolev inequalities for differential forms in Euclidean spaces, Sci. China Math. 62(6) (2019), 10291040. MR 3951879.
5.Baldi, A., Franchi, B. and Pansu, P., Duality and $L^{\infty }$ differential forms on Heisenberg groups, in preparation, 2020.
6.Baldi, A., Franchi, B. and Pansu, P., L 1 -Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups, Adv. Math. 366 (2020), 107084. MR 4070308.
7.Baldi, A., Franchi, B. and Pansu, P., Orlicz spaces and endpoint Sobolev–Poincaré inequalities for differential forms in Heisenberg groups, Matematiche (Catania) 75 (2020), 167194.
8.Baldi, A., Franchi, B., Tchou, N. and Tesi, M. C., Compensated compactness for differential forms in Carnot groups and applications, Adv. Math. 223(5) (2010), 15551607.
9.Baldi, A., Franchi, B. and Tesi, M. C., Compensated compactness in the contact complex of Heisenberg groups, Indiana Univ. Math. J. 57 (2008), 133186.
10.Baldi, A., Franchi, B. and Tesi, M. C., Hypoellipticity, fundamental solution and Liouville type theorem for matrix–valued differential operators in Carnot groups, J. Eur. Math. Soc. (JEMS) 11(4) (2009), 777798.
11.Baldi, A., Franchi, B. and Tripaldi, F., Gagliardo–Nirenberg inequalities for horizontal vector fields in the Engel group and in the seven-dimensional quaternionic Heisenberg group, in Geometric Methods in PDE’s, Springer INdAM Series, Volume 13, pp. 287312 (Springer, Cham, 2015). MR 3617226.
12.Balogh, Z. M., Fässler, K. and Peltonen, K., Uniformly quasiregular maps on the compactified Heisenberg group, J. Geom. Anal. 22(3) (2012), 633665. MR 2927672.
13.Bernig, A., Natural operations on differential forms on contact manifolds, Differential Geom. Appl. 50 (2017), 3451. MR 3588639.
14.Bonfiglioli, A., Lanconelli, E. and Uguzzoni, F., Stratified Lie groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics, (Springer, Berlin, 2007). MR 2363343.
15.Bourgain, J. and Brezis, Haïm, New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc. (JEMS) 9(2) (2007), 277315. MR 2293957  (2009h:35062).
16.Bryant, R., Eastwood, M., Rod Gover, A. and Neusser, K., Some differential complexes within and beyond parabolic geometry, Preprint, 2011, arXiv:1112.2142.
17.Capogna, L., Danielli, D. and Garofalo, N., Subelliptic mollifiers and a basic pointwise estimate of Poincaré type, Math. Z. 226(1) (1997), 147154. MR 1472145.
18.Chanillo, S. and Van Schaftingen, J., Subelliptic Bourgain-Brezis estimates on groups, Math. Res. Lett. 16(3) (2009), 487501. MR 2511628  (2010f:35042).
19.Fässler, K., Lukyanenko, A. and Peltonen, K., Quasiregular mappings on sub-Riemannian manifolds, J. Geom. Anal. 26(3) (2016), 17541794. MR 3511457.
20.Folland, G. B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13(2) (1975), 161207. MR 0494315 (58 #13215).
21.Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Groups, Mathematical Notes, Volume 28, (Princeton University Press, Princeton, NJ, 1982). MR 657581  (84h:43027).
22.Franchi, B., Gutiérrez, C. E. and Wheeden, R. L., Weighted Sobolev–Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations 19(3–4) (1994), 523604. MR 1265808.
23.Franchi, B., Lu, G. and Wheeden, R. L., Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier (Grenoble) 45(2) (1995), 577604. MR 1343563  (96i:46037).
24.Franchi, B., Lu, G. and Wheeden, R. L., A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type, Int. Math. Res. Not. IMRN (1) (1996), 114. MR 1383947  (97k:26012).
25.Franchi, B., Serapioni, R. and Cassano, F. S., Meyers–Serrin type theorems and relaxation of variational integrals depending on vector fields, Houston J. Math. 22(4) (1996), 859890. MR 1437714.
26.Franchi, B. and Serapioni, R. P., Intrinsic Lipschitz graphs within Carnot groups, J. Geom. Anal. 26(3) (2016), 19461994. MR 3511465.
27.Franchi, B., Serapioni, R. and Cassano, F. S., Regular submanifolds, graphs and area formula in Heisenberg groups, Adv. Math. 211(1) (2007), 152203. MR 2313532  (2008h:49030).
28.Gallot, S., Hulin, D. and Lafontaine, J., Riemannian Geometry, third edition, Universitext, (Springer, Berlin, 2004). MR 2088027.
29.Gol’dshtein, V. M., Kuz’minov, V. I. and Shvedov, I. A., L p-cohomology of warped cylinders, Sibirsk. Mat. Zh. 31(6) (1990), 5563. MR 1097955.
30.Gromov, M., Carnot–Carathéodory spaces seen from within, in Sub-Riemannian Geometry, Progress in Mathematics, Volume 144, pp. 79323 (Birkhäuser, Basel, 1996). MR 1421823  (2000f:53034).
31.Heinonen, J. and Holopainen, I., Quasiregular maps on Carnot groups, J. Geom. Anal. 7(1) (1997), 109148. MR 1630785.
32.Helffer, B. and Nourrigat, J., Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Progress in Mathematics, Volume 58, (Birkhäuser Boston Inc., Boston, MA, 1985). MR 897103  (88i:35029).
33.Iwaniec, T. and Lutoborski, A., Integral estimates for null Lagrangians, Arch. Ration. Mech. Anal. 125(1) (1993), 2579. MR 1241286  (95c:58054).
34.Jerison, D., The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J. 53(2) (1986), 503523. MR 850547  (87i:35027).
35.Kim, Y., Quasiconformal conjugacy classes of parabolic isometries of complex hyperbolic space, Pacific J. Math. 270(1) (2014), 129149. MR 3245851.
36.Lanzani, L. and Stein, E. M., A note on div curl inequalities, Math. Res. Lett. 12(1) (2005), 5761. MR 2122730  (2005m:58001).
37.Lukyanenko, A., Geometric mapping theory of the Heisenberg group, sub-Riemannian manifolds, and hyperbolic spaces, ProQuest LLC, Ann Arbor, MI, 2014, Ph.D. Thesis, University of Illinois at Urbana-Champaign. MR 3322035.
38.Maheux, P. and Saloff-Coste, L., Analyse sur les boules d’un opérateur sous-elliptique, Math. Ann. 303(4) (1995), 713740. MR 1359957  (96m:35049).
39.Martinet, J., Formes de contact sur les variétés de dimension 3, Lecture Notes in Mathematics, Volume 209, pp. 142163 (Springer, Berlin, 1971). MR 0350771.
40.Mattila, P., Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, Volume 44, (Cambridge University Press, Cambridge, 1995). MR 1333890.
41.McDuff, D. and Salamon, D., Introduction to Symplectic Topology, second edition, Oxford Mathematical Monographs, (The Clarendon Press, Oxford University Press, New York, 1998). MR 1698616.
42.Mitrea, D., Mitrea, M. and Monniaux, S., The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains, Commun. Pure Appl. Anal. 7(6) (2008), 12951333. MR 2425010.
43.Montgomery, R., A tour of Subriemannian Geometries, their Geodesics and Applications, Mathematical Surveys and Monographs, Volume 91, (American Mathematical Society, Providence, RI, 2002). MR 1867362  (2002m:53045).
44.Müller, D., Peloso, M. M. and Ricci, F., Analysis of the Hodge Laplacian on the Heisenberg group, Mem. Amer. Math. Soc. 233(1095) (2015), vi+91. MR 3289035.
45.Pansu, P., Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129(1) (1989), 160. MR 979599  (90e:53058).
46.Pansu, P., Cohomologie L p en degré 1 des espaces homogènes, Potential Anal. 27(2) (2007), 151165. MR 2322503.
47.Pansu, P., Cup-products in $l^{q,p}$-cohomology: discretization and quasi-isometry invariance, Preprint, 2017, arXiv:1702.04984.
48.Pansu, P. and Rumin, M., On the q, p cohomology of Carnot groups, Ann. H. Lebesgue 1 (2018), 267295. MR 3963292.
49.Pansu, P. and Tripaldi, F., Averages and the q, 1 cohomology of Heisenberg groups, Ann. Math. Blaise Pascal 26(1) (2019), 81100. (en).
50.Rumin, M., Formes différentielles sur les variétés de contact, J. Differential Geom. 39(2) (1994), 281330. MR 1267892  (95g:58221).
51.Rumin, M., Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups, C. R. Acad. Sci. Paris Sér. I Math. 329(11) (1999), 985990. MR 1733906  (2001g:53063).
52.Rumin, M., Sub-Riemannian limit of the differential form spectrum of contact manifolds, Geom. Funct. Anal. 10(2) (2000), 407452. MR 1771424  (2002f:53044).
53.Rumin, M., Around heat decay on forms and relations of nilpotent Lie groups, Séminaire de Théorie Spectrale et Géométrie, Volume 19, pp. 123164. MR 1909080  (2003f:58062).
54.Rumin, M., An introduction to spectral and differential geometry in Carnot–Carathéodory spaces, Rend. Circ. Mat. Palermo (2) Suppl. 75 (2005), 139196. MR 2152359  (2006g:58053).
55.Schwarz, G., Hodge Decomposition—A Method for Solving Boundary Value Problems, Lecture Notes in Mathematics, Volume 1607, (Springer, Berlin, 1995). MR 1367287  (96k:58222).
56.Stein, E. M., Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, Volume 43, (Princeton University Press, Princeton, NJ, 1993). With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. MR 1232192  (95c:42002).
57.Varopoulos, N. Th., Saloff-Coste, L. and Coulhon, T., Analysis and Geometry on Groups, Cambridge Tracts in Mathematics, Volume 100, (Cambridge University Press, Cambridge, 1992). MR 1218884  (95f:43008).
58.Warner, F. W., Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, Volume 94, (Springer, New York, 1983). Corrected reprint of the 1971 edition. MR 722297.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

POINCARÉ AND SOBOLEV INEQUALITIES FOR DIFFERENTIAL FORMS IN HEISENBERG GROUPS AND CONTACT MANIFOLDS

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.