Hardy-type inequalities related to degenerate elliptic differential operators
- [1] Dipartimento di Matematica Via E. Orabona, 4 I-70125 Bari, Italy
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)
- Volume: 4, Issue: 3, page 451-486
- ISSN: 0391-173X
Access Full Article
topAbstract
topHow to cite
topReferences
top- [1] Z. M. Balogh, I. Holopainen and J. T. Tyson, Singular solutions, homogeneous norms, and quasiconformal mappings in Carnot groups, Math. Ann. 324 (2002), 159–186. Zbl1014.22009MR1931762
- [2] Z. M. Balogh and J. T. Tyson, Polar coordinates in Carnot groups, Math. Z. 241 (2002), 697–730. Zbl1015.22005MR1942237
- [3] P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), 121–139. Zbl0556.35063MR742415
- [4] G. Barbatis, S. Filippas and A. Tertikas, Series expansion for Hardy inequalities, Indiana Univ. Math. J. 52 (2003), 171–190. Zbl1035.26014MR1970026
- [5] G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004), 2169–2196. Zbl1129.26019MR2048514
- [6] T. Bieske and J. Gong, The P-Laplace Equation on a class of Grushin-type Spaces, Proc. Amer. Math. Soc., to appear. Zbl1107.35007MR2240671
- [7] A. Bonfiglioli and F. Uguzzoni, A Note on Lifting of Carnot groups, Rev. Mat. Iberoamericana. To appear. Zbl1100.35029MR2232674
- [8] A. Bonfiglioli and F. Uguzzoni, Nonlinear Liouville theorems for some critical problems on H-type groups, J. Funct. Anal. 207 (2004), 161–215. Zbl1045.35018MR2027639
- [9] H. Brezis and X. Cabré, Some simple nonlinear PDE’s without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), 223–262. Zbl0907.35048MR1638143
- [10] H. Brezis and M. Marcus, Hardy’s inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 217–237 (1998). Dedicated to Ennio De Giorgi. Zbl1011.46027MR1655516
- [11] H. Brezis, M. Marcus and I. Shafrir, Extremal functions for Hardy’s inequality with weight, J. Funct. Anal. 171 (2000), 177–191. Zbl0953.26006MR1742864
- [12] H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), 443–469. Zbl0894.35038MR1605678
- [13] L. Capogna, Regularity for quasilinear equations and -quasiconformal maps in Carnot groups, Math. Ann. 313 (1999), no. 2, 263–295. Zbl0927.35024MR1679786
- [14] 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. Zbl0802.35024MR1239930
- [15] L. Capogna, D. Danielli and N. Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1996), 1153–1196. Zbl0878.35020MR1420920
- [16] G. Carron, Inégalités de Hardy sur les variétés riemanniennes non-compactes, J. Math. Pures Appl. (9) 76 (1997), 883–891. Zbl0886.58111MR1489943
- [17] J. Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc. 83 (1981), 69–70. Zbl0475.43010MR619983
- [18] L. D’Ambrosio, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc. 132 (2004), 725–734. Zbl1049.35077MR2019949
- [19] L. D’Ambrosio, Some Hardy Inequalities on the Heisenberg Group, Differential Equations 40 (2004), 552–564. Zbl1073.22003MR2153649
- [20] L. D’Ambrosio and S. Lucente, Nonlinear Liouville theorems for Grushin and Tricomi operators, J. Differential Equations 193 (2003), 511–541. Zbl1040.35012MR1998967
- [21] L. D’Ambrosio, E. Mitidieri and S. I. Pohozaev, Representation Formulae and Inequalities for Solutions od a Class of Second Order Partial Differential Equations, Trans. Amer. Math. Soc. (2005), PII S 0002-9947(05)03717-7. Zbl1081.35014MR2177044
- [22] D. Danielli, N. Garofalo and D.-M. Nhieu, Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003), 263–341. Zbl1077.22007MR2014879
- [23] E. B. Davies, The Hardy constant, Q. J. Math. (2) 46 (1995), 417–431. Zbl0857.26005MR1366614
- [24] E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in , Math. Z. 227 (1998), 511–523. Zbl0903.58049MR1612685
- [25] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161–207. Zbl0312.35026MR494315
- [26] G. B. Folland and E. M. Stein, “ Hardy spaces on homogeneous groups”, volume 28 of Mathematical Notes, Princeton University Press, Princeton, N.J., 1982. Zbl0508.42025MR657581
- [27] L. Gallardo, Capacités, mouvement brownien et problème de l’épine de Lebesgue sur les groupes de Lie nilpotents, In: “Probability measures on groups”, Oberwolfach, 1981, volume 928 of Lecture Notes in Math., Springer, Berlin, pp. 96–120. Zbl0483.60072MR669065
- [28] J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998), 441–476. Zbl0918.35052MR1616905
- [29] N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble) 40 (1990), 313–356. Zbl0694.22003MR1070830
- [30] N. Garofalo and D. Vassilev, Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type, Duke Math. J. 106 (2001), 411–448. Zbl1012.35014MR1813232
- [31] F. Gazzola, H.-C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc. 356 (2004), 2149–2168. Zbl1079.46021MR2048513
- [32] J. A. Goldstein and Q. S. Zhang, On a degenerate heat equation with a singular potential, J. Funct. Anal. 186 (2001), 342–359. Zbl1056.35093MR1864826
- [33] P. C. Greiner, A fundamental solution for a nonelliptic partial differential operator, Canad. J. Math. 31 (1979), 1107–1120. Zbl0475.35003MR546962
- [34] J. Heinonen, Calculus on Carnot groups, In: “Fall School in Analysis” (Jyväskylä, 1994), volume 68 of Report . Univ. Jyväskylä, Jyväskylä, pp. 1–31. Zbl0863.22009MR1351042
- [35] J. Heinonen and I. Holopainen, Quasiregular maps on Carnot groups, J. Geom. Anal. 7 (1997), 109–148. Zbl0905.30018MR1630785
- [36] A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147–153. Zbl0393.35015MR554324
- [37] I. Kombe, Nonlinear degenerate parabolic equations for Baouendi-Grushin operators (2004), preprint. Zbl1117.35039MR2226410
- [38] G. Lu, J. Manfredi and B. Stroffolini, Convex functions on the Heisenberg group, Calc. Var. Partial Differential Equations 19 (2004), 1–22. Zbl1072.49019MR2027845
- [39] V. Magnani, Lipschitz continuity, Aleksandrov theorem and characterizations for H-convex function (2003). URL http://cvgmt.sns.it/papers/mag03a, Preprint. Zbl1115.49004MR2208954
- [40] M. Marcus, V. J. Mizel and Y. Pinchover, On the best constant for Hardy’s inequality in , Trans. Amer. Math. Soc. 350 (1998), 3237–3255. Zbl0917.26016MR1458330
- [41] T. Matskewich and P. E. Sobolevskii, The best possible constant in generalized Hardy’s inequality for convex domain in , Nonlinear Anal. 28 (1997), 1601–1610. Zbl0876.46025MR1431208
- [42] T. Matskewich and P. E. Sobolevskii, The sharp constant in Hardy’s inequality for complement of bounded domain, Nonlinear Anal. 33 (1998), 105–120. Zbl0930.26009MR1621089
- [43] V. G. Maz’ja, “Sobolev spaces”, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin 1985. Translated from the Russian by T. O. Shaposhnikova. MR817985
- [44] E. Mitidieri, A simple approach to Hardy inequalities, Mat. Zametki 67 (2000), 563–572. Zbl0964.26010MR1769903
- [45] E. Mitidieri and S. I. Pohozaev, Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on , J. Evol. Equ. 1 (2001), 189–220. Zbl0988.35095MR1846746
- [46] E. Mitidieri and S. I. Pohozaev, A priori estimates and blow-up of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math. 234 (2001), 1–362. Zbl0987.35002MR1879326
- [47] P. Niu, H. Zhang and Y. Wang, Hardy-type and Rellich type inequalities on the Heisenberg group, Proc. Amer. Math. Soc. 129 (2001), 3623–3630. Zbl0979.35035MR1860496
- [48] S. Secchi, D. Smets and M. Willem, Remarks on a Hardy-Sobolev inequality, C. R. Math. Acad. Sci. Paris 336 (2003), 811–815. Zbl1035.35020MR1990020
- [49] J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal. 173 (2000), 103–153. Zbl0953.35053MR1760280
- [50] H. Zhang and P. Niu, Hardy-type inequalities and Pohozaev-type identities for a class of -degenerate subelliptic operators and applications, Nonlinear Anal. 54 (2003), 165–186. Zbl1033.47033MR1978971