Trace inequalities for Carnot-Carathéodory spaces and applications

Danielli, Donatella; Garofalo, Nicola; Nhieu, Duy-Minh

Danielli, Donatella ; Garofalo, Nicola ; Nhieu, Duy-Minh

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 27 (1998) no. 2, pp. 195-252

MR Zbl | 4 citations dans Numdam

@article{ASNSP_1998_4_27_2_195_0,
     author = {Danielli, Donatella and Garofalo, Nicola and Nhieu, Duy-Minh},
     title = {Trace inequalities for {Carnot-Carath\'eodory} spaces and applications},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {195--252},
     year = {1998},
     publisher = {Scuola normale superiore},
     volume = {Ser. 4, 27},
     number = {2},
     mrnumber = {1664688},
     zbl = {0938.46036},
     language = {en},
     url = {https://www.numdam.org/item/ASNSP_1998_4_27_2_195_0/}
}

TY  - JOUR
AU  - Danielli, Donatella
AU  - Garofalo, Nicola
AU  - Nhieu, Duy-Minh
TI  - Trace inequalities for Carnot-Carathéodory spaces and applications
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 1998
SP  - 195
EP  - 252
VL  - 27
IS  - 2
PB  - Scuola normale superiore
UR  - https://www.numdam.org/item/ASNSP_1998_4_27_2_195_0/
LA  - en
ID  - ASNSP_1998_4_27_2_195_0
ER  -

%0 Journal Article
%A Danielli, Donatella
%A Garofalo, Nicola
%A Nhieu, Duy-Minh
%T Trace inequalities for Carnot-Carathéodory spaces and applications
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 1998
%P 195-252
%V 27
%N 2
%I Scuola normale superiore
%U https://www.numdam.org/item/ASNSP_1998_4_27_2_195_0/
%G en
%F ASNSP_1998_4_27_2_195_0

Danielli, Donatella; Garofalo, Nicola; Nhieu, Duy-Minh. Trace inequalities for Carnot-Carathéodory spaces and applications. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 27 (1998) no. 2, pp. 195-252. https://www.numdam.org/item/ASNSP_1998_4_27_2_195_0/

Bibliographie
Cité par

[1] D. Adams, Traces ofpotentials arising from translation invariant operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1971), 203-217. | Zbl | MR | Numdam

[2] D. Adams, A trace inequality for generalized potentials, Studia Math. 48 (1973), 99-105. | Zbl | MR

[3] D. Adams, Weighted nonlinear potential theory, Trans. Amer. Math. Soc. 297 (1986), 73-94. | Zbl | MR

[4] D. Adams - L. Hedberg, "Function Spaces and Potential Theory", Springer-Verlag, 1996. | Zbl | MR

[5] M.S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math France 195 (1967), 45-87. | Zbl | MR | Numdam

[6] A. Bellaïche, "Sub-Riemannian Geometry", Birkhäuser, 1996. | MR

[7] M. Biroli - U. Mosco, A Saint- Venant type principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl. (IV) 169 (1995), 125-181. | Zbl | MR

[8] M. Biroli - U. Mosco, Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 6 (1995), 37-44. | Zbl | MR

[9] M. Biroli - U. Mosco, Sobolev inequalities on homogeneous spaces, Potential Anal. 4 (1995), 311-324. | Zbl | MR

[10] J. Boman, Lp estimates for very strongly elliptic systems, unpublished manuscript.

[11] S. BUCKLEY - P. KOSKELA - G. Lu (eds.), Boman equals John, In: "Proc. of the 16th Nevanlinna Coll". (Joensuu, 1995), de Gruyter, Berlin, 1996. | Zbl | MR

[12] H. Busemann, "Recent Synthetic Differential Geometry", Springer-Verlag, 1970. | Zbl | MR

[13] P. Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. 4 (1982), 213-230. | Zbl | MR | Numdam

[14] L. Capogna - D. Danielli - N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (1993), 1765-1794. | Zbl | MR

[15] L. Capogna - D. Danielli - N. Garofalo, An isoperimetric inequality and the geometric Sobolev embedding for vector fields, Comm. Anal. Geom. 2 (1994), 203-215. | Zbl | MR

[16] L. Capogna - D. Danielli - N. Garofalo, Subelliptic mollifiers and a basic pointwise estimate of Poincaré type, Math. Z. 226 (1997), 147-154. | Zbl | MR

[17] L. Capogna - D. Danielli - N. Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1997), 1153-1196. | Zbl | MR

[18] L. Capogna - N. Garofalo, Boundary behavior of nonegative solutions of subelliptic equations in NTA domains for Carnot-Carathéodory metrics, J. Fourier Anal. Appl. 4 (1998), to appear. | Zbl | MR

[19] L. Capogna - N. Garofalo - D.M. Nhieu, The Dirichlet problem for sub-Laplacians, preprint (1997).

[20] S.Y.A. Chang - J.M. Wilson - T.H. Wolff, Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv. 60 (1985), 217-246. | Zbl | MR

[21] S. Chanillo - R.L. Wheeden, Lp estimatesforfractional integrals and Sobolev inequalities with applications to Schrödinger operators, Comm. Partial Differential Equations 10 (1985), 1077-1166. | Zbl | MR

[22] F. Chiarenza, Regularity for solutions of quasilinear elliptic equations under minimal assumptions, Potential Anal. 4 (1995), 325-334. | Zbl | MR

[23] F. Chiarenza - M. Frasca, A remark on a paper by C. Fefferman, Proc. Amer. Math. Soc. 108 (1990), 407-409. | Zbl | MR

[24] I. Chavel, "Eigenvalues in Riemannian Geometry", Academic Press, Orlando, 1984. | Zbl | MR

[25] S. Cohn-Vossen, Existenz kürzester Wege, Dokl. Akad. Nauk SSSR 3 (1935), 339-342. | JFM

[26] R. Coifman - G. Weiss, "Analyse harmonique non-commutative sur certains espaces homogenes", Springer-Verlag, 1971. | Zbl | MR

[27] D. Danielli, Formules de représentation et théoreèmes d'inclusion pour des opérateurs sous-elliptiques, C.R. Acad. Sci.Paris Sér. I 314 (1992), 987-990. | Zbl | MR

[28] D. Danielli, A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations, Potential Analysis, to appear. | Zbl | MR

[29] G. David - S. Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs n. 38, American Mathematical Society, Providence, RI 1993. | Zbl | MR

[30] G. David - S. Semmes, Uniform rectifiability and singular sets, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 383-443. | Zbl | MR | Numdam

[31] L.C. Evans - R.F. Gariepy, "Measure Theory and Fine Properties of Functions", CRC press, 1992. | Zbl | MR

[32] H. Federer, "Geometric Measure Theory", Springer-Verlag, 1969. | Zbl | MR

[33] C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. 9 (1983), 129-206. | Zbl | MR

[34] C. FEFFERMAN - D. H. PHONG (eds.), Subelliptic eigenvalue problems, In: "Proceedings of the Conference in Harmonic Analysis in Honor of A. Zygmund", Wadsworth Math. Ser., Belmont, CA, 1981, pp. 530-606. | Zbl | MR

[35] C. Fefferman - A. Sanchez-Calle, Fundamental solutions for second order subelliptic operators, Ann. of Math. 124 (1986), 247-272. | Zbl | MR

[36] C. Fefferman - E. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. | Zbl | MR

[37] G.B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207. | Zbl | MR

[38] G.B. Folland - E.M. Stein, "Hardy Spaces on Homogeneous Groups", Princeton Univ. Press., 1982. | Zbl | MR

[39] B. Franchi, Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations, Trans. Amer. Math. Soc. 327 (1991), 125-158. | Zbl | MR

[40] B. Franchi - S. Gallot - R. Wheeden, Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann. 300 (1994), 557-571. | Zbl | MR

[41] B. Franchi - C. Gutiérrez - R. Wheeden, Weighted Sobolev-Poincare inequalities for Grushin type operators, Comm. Partial Differential Equations, 19 3-4 (1994), 523-604. | Zbl | MR

[42] B. Franchi - E. Lanconelli, Une metrique associeé à une classe d'operateurs elliptiques degénérés, Proceedings of the meeting "Linear Partial and Pseudo Differential Operators", Rend. Sem. Mat. Torino (1984), 105-114. | Zbl | MR

[43] B. Franchi - E. Lanconelli, Hölder regularity theorem for a class of linear non uniform elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), 523-451. | Zbl | MR | Numdam

[44] B. Franchi - E. Lanconelli, Une condition géométrique pour l'inégalité de Harnack, J. Math. Pures Appl. 64 (1985), 237-256. | Zbl | MR

[45] B. Franchi - G. Lu - R.L. Wheeden, Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier (Grenoble) 45 (1995), 577-604. | Zbl | MR | Numdam

[46] B. Franchi - G. Lu - R.L. Wheeden, A relationship between Poincaré type inequalities and representation formulas in spaces of homogeneous type, Internat Math. Res. Notices 1 (1996), 1-14. | Zbl | MR

[47] B. Franchi - R. Serapioni, Pointwise estimates for a class ofstrongly degenerate elliptic operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 527-568. | Zbl | MR | Numdam

[48] B. Franchi - R. Serapioni - F. Serra Cassano, Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields, preprint. | MR

[49] B. Franchi - R. Serapioni - F. Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associaed with Lipschitz continuous vector fields, Boll. Un. Mat. Ital. to appear. | Zbl | MR

[50] B. Franchi - R. Wheeden, Some remarks about Poincaré type inequalities and representation formulas in metric spaces of homogeneous type, preprint. | MR

[51] E. Gagliardo, Caratterizzazione delle tracce sula frontiera relative ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Univ. Padova 27 (1957), 284-305. | Zbl | MR | Numdam

[52] J. Garcia Cuerva - J.L. Rubio De Francia, "Weighted Norm Inequalities and Related Topics", North-Holland Mat. Stud. n. 116, 1985. | Zbl | MR

[53] N. Garofalo, "Recent Developments in the Theory of Subelliptic Equations and Its Geometric Aspects", Birkhäuser, to appear.

[54] N. Garofalo - 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. | Zbl | MR

[55] N. Garofalo - D.M. Nhieu, Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces,J. Analyse Math. 74 (1998), 67-97. | Zbl | MR

[56] M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems", Ann. of Math. Studies 105, Princeton Univ. Press, Princeton, N. J., 1983. | Zbl | MR

[57] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation", Birkhäuser, 1984. | Zbl | MR

[58] M. Gromov, "Structures métriques pour les variétés Riemanniennes" (rédigé par J. Lafontaine et P. Pansu), CEDIC ED., Paris, 1981. | Zbl | MR

[59] M. Gromov, Carnot-Carathéodory spaces seen from within, Inst. Hautes Études Sci. Publ. Math. (1994).

[60] V.V. Grushin, On a class of hypoelliptic operators, Math USSR-Sb., 12 3 (1970), 458-476. | Zbl

[61] P. Hajlasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403-415. | Zbl | MR

[62] P. Hajlasz - P. Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris Sér. I 320 (1995), 1211-1215. | Zbl | MR

[63] L. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510. | Zbl | MR

[64] L. Hedberg - T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 23 (1983), 161-187. | Zbl | MR | Numdam

[65] H. Hörmander, Hypoelliptic second-order differential equations, Acta Math. 119 (1967), 147-171. | Zbl | MR

[66] D. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, II, J. Funct. Anal. 43 (1981), 224-257. | Zbl | MR

[67] D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503-523. | Zbl | MR

[68] D. Jerison - C.E. Kenig, Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. Math. 46 (1982), 80-147. | Zbl | MR

[69] F. John, Rotation and strain, Comm. Pure Appl. Math. 14 (1961), 391-413. | Zbl | MR

[70] R. Kerman - E.T. Sawyer, The trace inequality and eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier (Grenoble) 36 (1986), 207-228. | Zbl | MR | Numdam

[71] A. Kufner - O. John - S. Fucik, "Function Spaces", Prague: Academia Pub. House of the Czechoslovak Academy of Sciences, 1977. | Zbl | MR

[72] G. Lieberman, Sharp form of estimates for subsolutions and supersolutions of quasilinear elliptic equations involving measures, Comm. Partial Differential Equations 18 (1993), 1191-1212. | Zbl | MR

[73] G. Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications, Rev. Mat. Iberoamericana 8 (1992), 367-439. | Zbl | MR

[74] G. Lu, The sharp Poincaré inequality for free vector fields : an endpoint result, Rev. Mat. Iberoamericana 18 (1994), 453-466. | Zbl | MR

[75] G. Lu, Embedding theorems on Campanato-Morrey spaces for vector fields and applications, C. R. Acad. Sci. Paris Sér. I 320 (1995), 429-434. | Zbl | MR

[76] G. Lu, Embedding theorems on Campanato-Morrey spaces for vector fields of Hörmander type, Approx. Theory Appl., to appear. | Zbl | MR

[77] G. Lu, Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations, Publ. Mat. 40 (1996), 301-329. | Zbl | MR

[78] P. Maheux - L. Saloff-Coste, Analyse sur les boules d'un op'erateur sous-elliptique, Math. Ann. 303 (1995), 713-740. | Zbl | MR

[79] P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability ", Cambridg studies in advanced Mathematics n. 44, Cambridge University Press, 1995. | Zbl | MR

[80] V.G. Mazýa, "Sobolev Spaces", Springer-Verlag, 1985. | Zbl | MR

[81] V.G. Mazýa - I.E. Verbitsky, Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multiplier, Ark. Mat. 33 (1995), 81-115. | Zbl | MR

[82] M. Mekias, "Restriction to Hypersurfaces of Non-isotropic Sobolev Spaces", M.I.T. Ph.D Thesis, 1993.

[83] A. Nagel - E.M. Stein - S. Wainger, Balls and metrics defined by vector fields I: basic properties, Acta Math. 155 (1985), 103-147. | Zbl | MR

[84] O.A. Oleinik - E.V. Radkevich, "Second Order Equations with Non-negative Characteristic Form", (Mathematical Analysis 1969), Moscow: Itogi Nauki, 1971 (Russian), English translation: Providence, R.I., Amer. Math. Soc., 1973.

[85] R.S. Phillips - L. Sarason, Elliptic-parabolic equations of the second order, J. Math. Mech. 17 (1967/8), 891-917. | Zbl | MR

[86] J.M. Rakotoson, Quasilinear equations and spaces of Campanato-Morrey type, Comm. Partial Differential Equations 16 (1991), 1155-1182. | Zbl | MR

[87] J.M. Rakotoson - W.P. Ziemer, Local behavior of solutions of quasilinear elliptic equations with general structure, Trans. Amer. Math. Soc. 319 (1990), 747-764. | Zbl | MR

[88] L. Saloff-Coste, Parabolic Harnack inequality for divergence-form second-order differential operators, Potential Anal. 4 (1995), 429-467. | Zbl | MR

[89] J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math. 111 (1964), 243-302. | Zbl | MR

[90] J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Math. 113 (1965), 219-240. | Zbl | MR

[91] G. Stampacchia, Problemi al contorno per equazioni di tipo ellittico a derivate parziali e questioni di calcolo delle variazioni connesse, Ann. Mat. Pura Appl. (4) 33 (1952), 211-238. | Zbl | MR

[92] E.M. Stein, "Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals", Princeton Univ. Press., 1993. | Zbl | MR

[93] Robert S. Strichartz, Sub-Riemannian geometry, J. Differential Geom. 24 (1986), 221-263. | Zbl | MR

[94] J. Strömberg - A. Torchinsky, "Weights, Sharp Maximal Functions and Hardy Spaces", Lecture Notes in Mathematics 1381, Springer-Verlag, 1989. | Zbl

[95] P. Tomter, Consant mean curvature surfaces in the Heisenberg group, Proc. Symp. Pure Math. 54 (1993), Part I, 485-495. | Zbl | MR

[96] N. Th. Varopoulos, Fonctions harmoniques sure les groupes de Lie, C.R. Acad. Sci. Paris Sér. I 304 (1987), 519-521. | Zbl | MR

[97] N. Th. Varopoulos - L. Saloff-Coste - T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics 100, Cambridge University press, 1992. | Zbl | MR

[98] S.K. Vodop'Yanov, Weighted Lp-potential theory on homogeneous groups, Sibirsk., Mat. Zh. 33 (1992), 29-48. | Zbl | MR

[99] C.J. Xu, Subelliptic variational problems, Bull. Soc. Math. France 118 (1990), 147-169. | Zbl | MR | Numdam

[100] P. Zamboni, Local behavior of solutions of quasilinear elliptic equations with coefficients in Morrey spaces, Rend. Mat. Appl. 15 (1995), 251-262. | Zbl | MR

[101] W.P. Ziemer, "Weakly Differentiable Functions", Springer-Verlag, 1989. | Zbl | MR