×

Trace inequalities for Carnot-Carathéodory spaces and applications. (English) Zbl 0938.46036

The authors are interested in traces of Sobolev spaces defined on domain in \(\mathbb{R}^n\) by the family of vector fields \(\{X_1,\dots, X_m\}\) with real-valued locally Lipschitz coefficients. The trace inequalities are formulated in terms of nonnegative Borel measures \(\mu\) on \(\mathbb{R}^n\). The classical example is a Borel measure given by the volume of a compact \(C^1\)-submanifold. The inequalities of the following types are investigated: \[ \begin{aligned} \Biggl(\int_{B(x,r)}|f- f_{B,\mu}|^q d\mu\Biggr)^{1/q} & \leq C\Biggl(\int_{B(x,\beta r)}|XF|^p dx\Biggr)^{1/p},\\ \Biggl(\int_{B(x,r)}|f|^p d\mu\Biggr)^{1/q} & \leq C\Biggl(\int_{B(x,\beta r)}|Xf|^p dx\Biggr)^{1/p},\end{aligned} \] where \(1\leq p\leq q<\infty\), \(\beta>1\), \(f_{B,\mu}= {1\over\mu(B)} \int_{B(x,r)}f d\mu\) and \(Xf= (X_1f,\dots, X_mf)\) is a gradient associated to the vector fields. It is assumed that the vector fields satisfy the following conditions:
(H.1) the metric Carnot-Carathéodory topology is equivalent to the Euclidean one,
(H.2) the Carnot-Carathéodory balls satisfy the doubling condition,
(H.3) the weak-\(L^1\) Poincaré type inequality holds for the gradient \(X\).
The following types of vector fields satisfy the assumptions (H.1)–(H.3): the Hörmander finite rank vector fields, the Baouendi-Grushkin vector fields, Lipschitz vector fields associated to the subelliptic operators.
Different assumptions concerning the domains and the Borel measures are regarded. In particular, bounded domains in Heisenberg groups with \(C^2\)-boundaries are considered. Several applications are mentioned.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
65H10 Numerical computation of solutions to systems of equations
58J55 Bifurcation theory for PDEs on manifolds
46N20 Applications of functional analysis to differential and integral equations
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] D. Adams , Traces ofpotentials arising from translation invariant operators , Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 ( 1971 ), 203 - 217 . Numdam | MR 287301 | Zbl 0219.46027 · Zbl 0219.46027
[2] D. Adams , A trace inequality for generalized potentials , Studia Math. 48 ( 1973 ), 99 - 105 . Article | MR 336316 | Zbl 0237.46037 · Zbl 0237.46037
[3] D. Adams , Weighted nonlinear potential theory , Trans. Amer. Math. Soc. 297 ( 1986 ), 73 - 94 . MR 849468 | Zbl 0656.31012 · Zbl 0656.31012 · doi:10.2307/2000457
[4] D. Adams - L. Hedberg , ” Function Spaces and Potential Theory ”, Springer-Verlag , 1996 . MR 1411441 | Zbl 0834.46021 · Zbl 0834.46021
[5] M.S. Baouendi , Sur une classe d’opérateurs elliptiques dégénérés , Bull. Soc. Math France 195 ( 1967 ), 45 - 87 . Numdam | MR 228819 | Zbl 0179.19501 · Zbl 0179.19501
[6] A. Bellaïche , ” Sub-Riemannian Geometry ”, Birkhäuser , 1996 . MR 1421821 · Zbl 0862.53031
[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 . MR 1378473 | Zbl 0851.31008 · Zbl 0851.31008 · doi:10.1007/BF01759352
[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 . MR 1340280 | Zbl 0837.31006 · Zbl 0837.31006
[9] M. Biroli - U. Mosco , Sobolev inequalities on homogeneous spaces , Potential Anal. 4 ( 1995 ), 311 - 324 . MR 1354886 | Zbl 0833.46020 · Zbl 0833.46020 · doi:10.1007/BF01053449
[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 . MR 1427074 | Zbl 0861.43007 · Zbl 0861.43007
[12] H. Busemann , ” Recent Synthetic Differential Geometry ”, Springer-Verlag , 1970 . MR 296877 | Zbl 0194.53701 · Zbl 0194.53701
[13] P. Buser , A note on the isoperimetric constant , Ann. Sci. École Norm. Sup. 4 ( 1982 ), 213 - 230 . Numdam | MR 683635 | Zbl 0501.53030 · Zbl 0501.53030
[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 . MR 1239930 | Zbl 0802.35024 · Zbl 0802.35024 · doi:10.1080/03605309308820992
[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 . MR 1312686 | Zbl 0864.46018 · Zbl 0864.46018
[16] L. Capogna - D. Danielli - N. Garofalo , Subelliptic mollifiers and a basic pointwise estimate of Poincaré type , Math. Z. 226 ( 1997 ), 147 - 154 . MR 1472145 | Zbl 0893.35023 · Zbl 0893.35023 · doi:10.1007/PL00004330
[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 . MR 1420920 | Zbl 0878.35020 · Zbl 0878.35020 · doi:10.1353/ajm.1996.0046
[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. MR 1658616 | Zbl 0926.35043 · Zbl 0926.35043 · doi:10.1007/BF02498217
[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 . MR 800004 | Zbl 0575.42025 · Zbl 0575.42025 · doi:10.1007/BF02567411
[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 . MR 806256 | Zbl 0578.46024 · Zbl 0578.46024 · doi:10.1080/03605308508820401
[22] F. Chiarenza , Regularity for solutions of quasilinear elliptic equations under minimal assumptions , Potential Anal. 4 ( 1995 ), 325 - 334 . MR 1354887 | Zbl 0838.35022 · Zbl 0838.35022 · doi:10.1007/BF01053450
[23] F. Chiarenza - M. Frasca , A remark on a paper by C. Fefferman , Proc. Amer. Math. Soc. 108 ( 1990 ), 407 - 409 . MR 1027825 | Zbl 0694.46029 · Zbl 0694.46029 · doi:10.2307/2048289
[24] I. Chavel , ” Eigenvalues in Riemannian Geometry ”, Academic Press , Orlando , 1984 . MR 768584 | Zbl 0551.53001 · Zbl 0551.53001
[25] S. Cohn-Vossen , Existenz kürzester Wege , Dokl. Akad. Nauk SSSR 3 ( 1935 ), 339 - 342 . JFM 61.1439.01 · JFM 61.1439.01
[26] R. Coifman - G. Weiss , ”Analyse harmonique non-commutative sur certains espaces homogenes” , Springer-Verlag , 1971 . MR 499948 | Zbl 0224.43006 · Zbl 0224.43006 · doi:10.1007/BFb0058946
[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 . MR 1168522 | Zbl 0768.46019 · Zbl 0768.46019
[28] D. Danielli , A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations , Potential Analysis , to appear. MR 1719837 | Zbl 0940.35057 · Zbl 0940.35057 · doi:10.1023/A:1008674906902
[29] G. David - S. Semmes , Analysis of and on uniformly rectifiable sets , Mathematical Surveys and Monographs n. 38 , American Mathematical Society , Providence, RI 1993 . MR 1251061 | Zbl 0832.42008 · Zbl 0832.42008
[30] G. David - S. Semmes , Uniform rectifiability and singular sets , Ann. Inst. H. Poincaré Anal. Non Linéaire 13 ( 1996 ), 383 - 443 . Numdam | MR 1404317 | Zbl 0908.49030 · Zbl 0908.49030
[31] L.C. Evans - R.F. Gariepy , ” Measure Theory and Fine Properties of Functions ”, CRC press , 1992 . MR 1158660 | Zbl 0804.28001 · Zbl 0804.28001
[32] H. Federer , ” Geometric Measure Theory ”, Springer-Verlag , 1969 . MR 257325 | Zbl 0176.00801 · Zbl 0176.00801
[33] C. Fefferman , The uncertainty principle , Bull. Amer. Math. Soc. 9 ( 1983 ), 129 - 206 . Article | MR 707957 | Zbl 0526.35080 · Zbl 0526.35080 · doi:10.1090/S0273-0979-1983-15154-6
[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 . MR 730094 | Zbl 0503.35071 · Zbl 0503.35071
[35] C. Fefferman - A. Sanchez-Calle , Fundamental solutions for second order subelliptic operators , Ann. of Math. 124 ( 1986 ), 247 - 272 . MR 855295 | Zbl 0613.35002 · Zbl 0613.35002 · doi:10.2307/1971278
[36] C. Fefferman - E. Stein , Some maximal inequalities , Amer. J. Math. 93 ( 1971 ), 107 - 115 . MR 284802 | Zbl 0222.26019 · Zbl 0222.26019 · doi:10.2307/2373450
[37] G.B. Folland , Subelliptic estimates and function spaces on nilpotent Lie groups , Ark. Mat. 13 ( 1975 ), 161 - 207 . MR 494315 | Zbl 0312.35026 · Zbl 0312.35026 · doi:10.1007/BF02386204
[38] G.B. Folland - E.M. Stein , ” Hardy Spaces on Homogeneous Groups ”, Princeton Univ. Press. , 1982 . MR 657581 | Zbl 0508.42025 · Zbl 0508.42025
[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 . MR 1040042 | Zbl 0751.46023 · Zbl 0751.46023 · doi:10.2307/2001837
[40] B. Franchi - S. Gallot - R. Wheeden , Sobolev and isoperimetric inequalities for degenerate metrics , Math. Ann. 300 ( 1994 ), 557 - 571 . MR 1314734 | Zbl 0830.46027 · Zbl 0830.46027 · doi:10.1007/BF01450501
[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 . MR 1265808 | Zbl 0822.46032 · Zbl 0822.46032 · doi:10.1080/03605309408821025
[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 . MR 745979 | Zbl 0553.35033 · Zbl 0553.35033
[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 . Numdam | MR 753153 | Zbl 0552.35032 · Zbl 0552.35032
[44] B. Franchi - E. Lanconelli , Une condition géométrique pour l’inégalité de Harnack , J. Math. Pures Appl. 64 ( 1985 ), 237 - 256 . MR 823405 | Zbl 0599.35134 · Zbl 0599.35134
[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 . Numdam | MR 1343563 | Zbl 0820.46026 · Zbl 0820.46026 · doi:10.5802/aif.1466
[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 . MR 1383947 | Zbl 0856.43006 · Zbl 0856.43006 · doi:10.1155/S1073792896000013
[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 . Numdam | MR 963489 | Zbl 0685.35046 · Zbl 0685.35046
[48] B. Franchi - R. Serapioni - F. Serra Cassano , Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields , preprint. MR 1437714 · Zbl 0876.49014
[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. MR 1448000 | Zbl 0952.49010 · Zbl 0952.49010
[50] B. Franchi - R. Wheeden , Some remarks about Poincaré type inequalities and representation formulas in metric spaces of homogeneous type, preprint . MR 1731670
[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 . Numdam | MR 102739 | Zbl 0087.10902 · Zbl 0087.10902
[52] J. Garcia Cuerva - J.L. Rubio De Francia , ” Weighted Norm Inequalities and Related Topics ”, North-Holland Mat. Stud. n. 116 , 1985 . MR 807149 | Zbl 0578.46046 · Zbl 0578.46046
[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 . MR 1404326 | Zbl 0880.35032 · Zbl 0880.35032 · doi:10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A
[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 . MR 1631642 | Zbl 0906.46026 · Zbl 0906.46026 · doi:10.1007/BF02819446
[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 . MR 717034 | Zbl 0516.49003 · Zbl 0516.49003
[57] E. Giusti , ” Minimal Surfaces and Functions of Bounded Variation ”, Birkhäuser , 1984 . MR 775682 | Zbl 0545.49018 · Zbl 0545.49018
[58] M. Gromov , ” Structures métriques pour les variétés Riemanniennes ” (rédigé par J. Lafontaine et P. Pansu), CEDIC ED ., Paris , 1981 . MR 682063 | Zbl 0509.53034 · Zbl 0509.53034
[59] M. Gromov , Carnot-Carathéodory spaces seen from within , Inst. Hautes Études Sci. Publ. Math. ( 1994 ). · Zbl 0864.53025
[60] V.V. Grushin , On a class of hypoelliptic operators , Math USSR-Sb. , 12 3 ( 1970 ), 458 - 476 . Zbl 0252.35057 · Zbl 0252.35057 · doi:10.1070/SM1970v012n03ABEH000931
[61] P. Hajlasz , Sobolev spaces on an arbitrary metric space , Potential Anal. 5 ( 1996 ), 403 - 415 . MR 1401074 | Zbl 0859.46022 · Zbl 0859.46022
[62] P. Hajlasz - P. Koskela , Sobolev meets Poincaré , C. R. Acad. Sci. Paris Sér. I 320 ( 1995 ), 1211 - 1215 . MR 1336257 | Zbl 0837.46024 · Zbl 0837.46024
[63] L. Hedberg , On certain convolution inequalities , Proc. Amer. Math. Soc. 36 ( 1972 ), 505 - 510 . MR 312232 | Zbl 0283.26003 · Zbl 0283.26003 · doi:10.2307/2039187
[64] L. Hedberg - T. Wolff , Thin sets in nonlinear potential theory , Ann. Inst. Fourier ( Grenoble ) 23 ( 1983 ), 161 - 187 . Numdam | MR 727526 | Zbl 0508.31008 · Zbl 0508.31008 · doi:10.5802/aif.944
[65] H. Hörmander , Hypoelliptic second-order differential equations , Acta Math. 119 ( 1967 ), 147 - 171 . MR 222474 | Zbl 0156.10701 · Zbl 0156.10701 · doi:10.1007/BF02392081
[66] D. Jerison , The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, II , J. Funct. Anal. 43 ( 1981 ), 224 - 257 . MR 633978 | Zbl 0493.58022 · Zbl 0493.58022 · doi:10.1016/0022-1236(81)90031-8
[67] D. Jerison , The Poincaré inequality for vector fields satisfying Hörmander’s condition , Duke Math. J. 53 ( 1986 ), 503 - 523 . Article | MR 850547 | Zbl 0614.35066 · Zbl 0614.35066 · doi:10.1215/S0012-7094-86-05329-9
[68] D. Jerison - C.E. Kenig , Boundary behavior of harmonic functions in non-tangentially accessible domains , Adv. Math. 46 ( 1982 ), 80 - 147 . MR 676988 | Zbl 0514.31003 · Zbl 0514.31003 · doi:10.1016/0001-8708(82)90055-X
[69] F. John , Rotation and strain , Comm. Pure Appl. Math. 14 ( 1961 ), 391 - 413 . MR 138225 | Zbl 0102.17404 · Zbl 0102.17404 · doi:10.1002/cpa.3160140316
[70] R. Kerman - E.T. Sawyer , The trace inequality and eigenvalue estimates for Schrödinger operators , Ann. Inst. Fourier ( Grenoble ) 36 ( 1986 ), 207 - 228 . Numdam | MR 867921 | Zbl 0591.47037 · Zbl 0591.47037 · doi:10.5802/aif.1074
[71] A. Kufner - O. John - S. Fucik , ” Function Spaces ”, Prague : Academia Pub. House of the Czechoslovak Academy of Sciences , 1977 . MR 482102 | Zbl 0364.46022 · Zbl 0364.46022
[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 . MR 1233190 | Zbl 0802.35041 · Zbl 0802.35041 · doi:10.1080/03605309308820969
[73] G. Lu , Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications , Rev. Mat. Iberoamericana 8 ( 1992 ), 367 - 439 . MR 1202416 | Zbl 0804.35015 · Zbl 0804.35015 · doi:10.4171/RMI/129
[74] G. Lu , The sharp Poincaré inequality for free vector fields : an endpoint result , Rev. Mat. Iberoamericana 18 ( 1994 ), 453 - 466 . MR 1286482 | Zbl 0860.35006 · Zbl 0860.35006 · doi:10.4171/RMI/158
[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 . MR 1320116 | Zbl 0842.46019 · Zbl 0842.46019
[76] G. Lu , Embedding theorems on Campanato-Morrey spaces for vector fields of Hörmander type , Approx. Theory Appl. , to appear. MR 1651473 | Zbl 0916.46026 · Zbl 0916.46026
[77] G. Lu , Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations , Publ. Mat. 40 ( 1996 ), 301 - 329 . MR 1425620 | Zbl 0873.35006 · Zbl 0873.35006 · doi:10.5565/PUBLMAT_40296_04
[78] P. Maheux - L. Saloff-Coste , Analyse sur les boules d’un op’erateur sous-elliptique , Math. Ann. 303 ( 1995 ), 713 - 740 . MR 1359957 | Zbl 0836.35106 · Zbl 0836.35106 · doi:10.1007/BF01461013
[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 . MR 1333890 | Zbl 0819.28004 · Zbl 0819.28004
[80] V.G. Mazýa , ” Sobolev Spaces ”, Springer-Verlag , 1985 . MR 817985 | Zbl 0692.46023 · Zbl 0692.46023
[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 . MR 1340271 | Zbl 0834.31006 · Zbl 0834.31006 · doi:10.1007/BF02559606
[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 . MR 793239 | Zbl 0578.32044 · Zbl 0578.32044 · doi:10.1007/BF02392539
[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 . MR 219868 | Zbl 0163.34402 · Zbl 0163.34402
[86] J.M. Rakotoson , Quasilinear equations and spaces of Campanato-Morrey type , Comm. Partial Differential Equations 16 ( 1991 ), 1155 - 1182 . MR 1116857 | Zbl 0827.35021 · Zbl 0827.35021 · doi:10.1080/03605309108820793
[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 . MR 998128 | Zbl 0708.35023 · Zbl 0708.35023 · doi:10.2307/2001263
[88] L. Saloff-Coste , Parabolic Harnack inequality for divergence-form second-order differential operators , Potential Anal. 4 ( 1995 ), 429 - 467 . MR 1354894 | Zbl 0840.31006 · Zbl 0840.31006 · doi:10.1007/BF01053457
[89] J. Serrin , Local behavior of solutions of quasilinear equations , Acta Math. 111 ( 1964 ), 243 - 302 . MR 170096 | Zbl 0128.09101 · Zbl 0128.09101 · doi:10.1007/BF02391014
[90] J. Serrin , Isolated singularities of solutions of quasilinear equations , Acta Math. 113 ( 1965 ), 219 - 240 . MR 176219 | Zbl 0173.39202 · Zbl 0173.39202 · doi:10.1007/BF02391778
[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 . MR 51451 | Zbl 0047.33902 · Zbl 0047.33902 · doi:10.1007/BF02418184
[92] E.M. Stein , ” Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals ”, Princeton Univ. Press. , 1993 . MR 1232192 | Zbl 0821.42001 · Zbl 0821.42001
[93] Robert S. Strichartz , Sub-Riemannian geometry , J. Differential Geom. 24 ( 1986 ), 221 - 263 . MR 862049 | Zbl 0609.53021 · Zbl 0609.53021
[94] J. Strömberg - A. Torchinsky , ” Weights, Sharp Maximal Functions and Hardy Spaces ”, Lecture Notes in Mathematics 1381 , Springer-Verlag , 1989 . Zbl 0676.42021 · Zbl 0676.42021 · doi:10.1007/BFb0091154
[95] P. Tomter , Consant mean curvature surfaces in the Heisenberg group , Proc. Symp. Pure Math. 54 ( 1993 ), Part I, 485 - 495 . MR 1216601 | Zbl 0799.53073 · Zbl 0799.53073
[96] N. Th. Varopoulos , Fonctions harmoniques sure les groupes de Lie , C.R. Acad. Sci. Paris Sér. I 304 ( 1987 ), 519 - 521 . MR 892879 | Zbl 0614.22002 · Zbl 0614.22002
[97] N. Th. Varopoulos - L. Saloff-Coste - T. Coulhon , Analysis and geometry on groups , Cambridge Tracts in Mathematics 100 , Cambridge University press , 1992 . MR 1218884 | Zbl 0813.22003 · Zbl 0813.22003
[98] S.K. Vodop’yanov , Weighted Lp-potential theory on homogeneous groups, Sibirsk. , Mat. Zh. 33 ( 1992 ), 29 - 48 . MR 1174058 | Zbl 0774.31009 · Zbl 0774.31009
[99] C.J. Xu , Subelliptic variational problems , Bull. Soc. Math. France 118 ( 1990 ), 147 - 169 . Numdam | MR 1087376 | Zbl 0717.49004 · Zbl 0717.49004
[100] P. Zamboni , Local behavior of solutions of quasilinear elliptic equations with coefficients in Morrey spaces , Rend. Mat. Appl. 15 ( 1995 ), 251 - 262 . MR 1339243 | Zbl 0832.35046 · Zbl 0832.35046
[101] W.P. Ziemer , ” Weakly Differentiable Functions ”, Springer-Verlag , 1989 . MR 1014685 | Zbl 0692.46022 · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.