Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields

Studia Mathematica

Volumn 139, Issue 3, 2000, Pages 213-244

  Morbidelli, Daniele ^a  

^a UNIVERSITY OF BOLOGNA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034360498     PISSN: 00393223     EISSN: None     Source Type: Journal    
DOI: 10.4064/sm-139-3-213-244     Document Type: Article

References (56)

1. D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995), 1033-1074.
2. G. Ben Arous and M. Gradinaru, Singularities of hypoelliptic Green functions, Potential Anal. 8 (1998), 217-258.
3. S. Berhanu and I. Pesenson, The trace problem for vector fields satisfying Hörmander's condition, Math. Z. 231 (1999), 103-122.
4. M. Biroli and U. Mosco, Sobolev inequalities on homogeneous spaces, Potential Anal. 4 (1995), 311-324.
5. J. M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), no. 1, 277-304.
6. S. Buckley, P. Koskela and G. Lu, Subelliptic Poincaré estimâtes: the case p < 1, Publ. Math. 39 (1995), 313-334.
7. L. Capogna, D. Danielli and N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (1993), 1765-1794.
8. _, _, _, The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. Anal. Geom. 2 (1994), 203-215.
9. _, _, _, Subelliptic mollifiers and a basic pointwise estimate of Poincaré type, Math. Z. 226 (1997), 147-154.
10. _, _, _, Capacitary estimates and the local behavior of solutions to nonlinear subelliptic equations, Amer. J. Math. 118 (1996), 1153-1196.
11. J. Y. Chemin et C. J. Xu, Inclusions de Sobolev en calcul de Weyl-Hörmander et champs de vecteurs sous-elliptiques, Ann. Sci. Ecole Norm. Sup. (4) 30 (1997), 719-751.
12. V. M. Chernikov and S. K. Vodop'yanov, Sobolev spaces and hypoelliptic equations I, II, Siberian Adv. Math. 6 (1996), no. 3, 27-67, and no. 4, 64-96.
13. W. L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1940), 98-105.
14. G. Citti and G. Di Fazio, Hölder continuity of the solutions for operators which are sum of squares of vector fields plus a potential, Proc. Amer. Math. Soc. 122 (1994), 741-750.
15. G. Citti, N. Garofalo and E. Lanconelli, Harnack's inequality for sum of squares of vector fields plus a potential, Amer. J. Math. 115 (1993), 699-734.
16. D. Danielli, A compact embedding theorem for a class of degenerate Sobolev spaces, Rend. Sem. Mat. Univ. Politec. Torino 49 (1991), 339-420.
17. C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, in: Proc. Conf. on Harmonic Analysis in honor of Antoni Zygmund, Wadsworth, Belmont, CA, 1983, 590-606.
18. C. Fefferman and A. Sánchez-Calle, Fundamental solution for second order subelliptic operators, Ann. of Math. 124 (1986), 247-272.
19. G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207.
20. B. Franchi, S. Gallot and R. Wheeden, Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann. 300 (1994), 557-571.
21. B. Franchi et E. Lanconelli, Une métrique associée à une classe d'opérateurs elliptiques dégénérés, in: Conference on Linear Partial and Pseudodifferential Operators, Rend. Sem. Mat. Univ. Politec. Torino (1983) (special issue), 105-114.
22. _, _, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa 10 (1983), 523-541.
23. _, _, An embedding theorem for Sobolev spaces related to non-smooth vector fields and Harnack inequality, Comm. Partial Differential Equations 9 (1984), 1237-1264.
24. B. Franchi, G. Lu and R. Wheeden, Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier (Grenoble) 45 (1995), 577-604.
25. _, _, _, A relationship between Poincaré type inequalities and representation formulas in spaces of homogeneous type, Internat. Math. Res. Notices 1996, no. 1, 1-14.
26. B. Franchi, R. Serapioni and F. Serra Cassano, Approximation and embedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields, Boll. Un. Mat. Ital. B (7) 11 (1997), 83-117.
27. N. Garofalo and D. M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), 1081-1144.
28. _, _, Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces, J. Anal. Math. 74 (1998), 67-97.
29. P. Hajlasz and P. Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris 320 (1995), 1211-1215.
30. _, _, Sobolev met Poincaré, Mem. Amer. Math. Soc. (to appear).
31. P. Hajlasz and P. Strzelecki, Subelliptic p-harmonic maps into spheres and the ghost of Hardy spaces, Math. Ann. 312 (1998), 341-362.
32. G. Hochschild, La structure des groupes de Lie, Dunod, Paris, 1968.
33. L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171.
34. L. Hörmander and A. Melin, Free systems of vector fields, Ark. Mat. 16 (1978), 83-88.
35. D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503-523.
36. D. Jerison and A. Sánchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J. 35 (1986), 835-854.
37. _, _, Subelliptic second order differential operators, in: Lecture Notes in Math. 1277, Springer, 1987, 46-77.
38. p estimates for elliptic equations under general Hörmander's condition, Topol. Methods Nonlinear Anal. 9 (1997), 249-258.
39. S. Kusuoka and D. W. Strook, Applications of the Malliavin calculus, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 34 (1987), 392-442.
40. _, _, Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator, Ann. of Math. 127 (1988), 165-189.
41. E. Lanconelli, Stime subellittiche e metriche Riemanniane singolari, Seminario di Analisi Matematica, Dipartimento di Matematica, Università di Bologna, A. A. 1982-83.
42. E. Lanconelli and D. Morbidelli, On the Poincaré inequality for vector fields, Ark. Mat. (to appear).
43. G. Lu, Existence and size estimates for the Green's functions of differential operators constructed from degenerate vector fields, Comm. Partial Differential Equations 17 (1992), 1213-1251.
44. _, Embedding theorems into Lipschitz and BMO spaces and applications to quasi-linear subelliptic differential equations, Publ. Math. 40 (1996), 301-329.
45. _, A note on a Poincaré type inequality for solutions to subelliptic equations, Comm. Partial Differential Equations 21 (1996), 235-254.
46. P. Maheux et L. Saloff-Coste, Analyse sur les boules d'un opérateur souselliptique, Math. Ann. 303 (1995), 713-746.
47. A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields I: basic properties, Acta Math. 155 (1985), 103-147.
48. W. C. Rheinboldt, Local mapping relations and global implicit function theorems, Trans. Amer. Math. Soc. 138 (1969), 183-198.
49. L. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320.
50. K. Saka, Besov spaces and Sobolev spaces on a nilpotent Lie group, Tôhoku Math. J. 31 (1979), 383-437.
51. L. Saloff-Coste, A note on Poincaré, Sobolev and Harnack inequalities, Internat. Math. Res. Notices 1992, no. 2, 27-38.
52. A. Sánchez-Calle, Fundamental solutions and geometry of sum of squares of vector fields, Invent. Math. 78 (1984), 143-160.
53. J.-P. Serre, Lie Algebras and Lie Groups, Lecture Notes in Math. 1500, Springer, 1992.
54. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge Univ. Press, 1992.
55. S. K. Vodop'yanov and I. G. Markina, Exceptional sets for solutions of subeliptic equations, Siberian Math. J. 36 (1995), 694-706.
56. C. J. Xu, Regularity for quasi linear second order subelliptic equations, Comm. Pure Appl. Math. 45 (1992), 77-96.