We prove a simple sufficient criterion to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second order differential operator . Namely, if ρ is a nonnegative weight such that , then the Hardy inequality
Keywords: Hardy inequality, Riemannian manifolds, Parabolic manifolds, Caccioppoli inequality, Weighted Gagliardo–Nirenberg inequality, Interpolation inequality
@article{AIHPC_2014__31_3_449_0, author = {D'Ambrosio, Lorenzo and Dipierro, Serena}, title = {Hardy inequalities on {Riemannian} manifolds and applications}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {449--475}, publisher = {Elsevier}, volume = {31}, number = {3}, year = {2014}, doi = {10.1016/j.anihpc.2013.04.004}, mrnumber = {3208450}, zbl = {1317.46022}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.04.004/} }
TY - JOUR AU - D'Ambrosio, Lorenzo AU - Dipierro, Serena TI - Hardy inequalities on Riemannian manifolds and applications JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 449 EP - 475 VL - 31 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.04.004/ DO - 10.1016/j.anihpc.2013.04.004 LA - en ID - AIHPC_2014__31_3_449_0 ER -
%0 Journal Article %A D'Ambrosio, Lorenzo %A Dipierro, Serena %T Hardy inequalities on Riemannian manifolds and applications %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 449-475 %V 31 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.04.004/ %R 10.1016/j.anihpc.2013.04.004 %G en %F AIHPC_2014__31_3_449_0
D'Ambrosio, Lorenzo; Dipierro, Serena. Hardy inequalities on Riemannian manifolds and applications. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, pp. 449-475. doi : 10.1016/j.anihpc.2013.04.004. http://www.numdam.org/articles/10.1016/j.anihpc.2013.04.004/
[1] Role of the fundamental solution in Hardy–Sobolev-type inequalities, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 1111 -1130 | MR | Zbl
, ,[2] Hardy type inequalities on complete Riemannian manifolds, Monatsh. Math. 163 (2011), 115 -129 | MR | Zbl
, ,[3] Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, Berlin (1998) | MR | Zbl
,[4] The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), 121 -139 | MR | Zbl
, ,[5] A unified approach to improved Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004), 2169 -2196 | MR | Zbl
, , ,[6] Hardy inequality and heat semigroup estimates for Riemannian manifolds with singular data, Comm. Partial Differential Equations 37 (2012), 885 -900 | MR | Zbl
, , , ,[7] Conformal Killing vector fields and Rellich type identities on Riemannian manifolds, I, Lect. Notes Semin. Interdiscip. Mat. vol. 7 (2008), 65 -80 | MR | Zbl
, ,[8] Conformal Killing vector fields and Rellich type identities on Riemannian manifolds, II, Mediterr. J. Math. 9 (2012), 1 -20 | MR | Zbl
, ,[9] Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8 no. 1 (1998), 223 -262 | EuDML | MR | Zbl
, ,[10] Hardy's inequalities revisited, Ann. Sc. Norm. Super. Cl. Sci. (4) 25 (1997), 217 -237 | EuDML | Numdam | MR | Zbl
, ,[11] Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madr. 10 (1997), 443 -469 | EuDML | MR | Zbl
, ,[12] Spectral stability of the Neumann Laplacian, J. Differential Equations 186 (2002), 485 -508 | MR | Zbl
, ,[13] First order interpolation inequalities with weights, Compos. Math. 53 no. 3 (1984), 259 -275 | EuDML | Numdam | MR | Zbl
, , ,[14] Complete manifolds with non-negative Ricci curvature and the Caffarelli–Kohn–Nirenberg inequalities, Compos. Math. 140 (2004), 818 -826 | MR | Zbl
, ,[15] Inégalités isopérimétriques de Faber–Krahn et conséquences, Actes de la Table Ronde de Géométrie Différentielle, Luminy, 1992, Semin. Congr. vol. 1 (1996), 205 -232 | MR | Zbl
,[16] Inégalités de Hardy sur les variétés Riemanniennes non-compactes, J. Math. Pures Appl. (9) 76 (1997), 883 -891 | MR
,[17] On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96 (1972), 413 -443 | MR | Zbl
, ,[18] Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc. 132 (2004), 725 -734 | MR | Zbl
,[19] Some Hardy inequalities on the Heisenberg group, Differ. Equ. 40 (2004), 552 -564 | MR | Zbl
,[20] Hardy-type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) IV (2005), 451 -586 | EuDML | Numdam | MR | Zbl
,[21] Explicit constants for Rellich inequalities in , Math. Z. 227 (1998), 511 -523 | MR | Zbl
, ,[22] Heat kernel and Hardy estimates for locally Euclidean manifolds with fractal boundaries, Geom. Funct. Anal. 3 (1993), 527 -563 | EuDML | MR | Zbl
, ,[23] Optimal Hardy-type inequality for second-order elliptic operator and applications, arXiv:1208.2342 | MR
, , ,[24] Optimal Hardy-type inequalities for elliptic operators, C. R. Acad. Sci. Paris 350 (2012), 475 -479 | MR | Zbl
, , ,[25] Some weighted Gagliardo–Nirenberg inequalities and applications, Proc. Amer. Math. Soc. 135 (2007), 2795 -2802 | MR | Zbl
, ,[26] The spectrum of the Laplacian of manifolds of positive curvature, Duke Math. J. 65 (1992), 1 -21 | MR | Zbl
, ,[27] Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc. 356 (2004), 2149 -2168 | MR | Zbl
, , ,[28] On the existence of a Green function on a manifold, Russian Math. Surveys 38 no. 1 (1983), 190 -191 | Zbl
,[29] On the existence of positive fundamental solutions of the Laplace equation on Riemannian manifold, Math. USSR Sb. 56 no. 2 (1987), 349 -357 | MR | Zbl
,[30] Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 36 no. 2 (1999), 135 -249 | MR | Zbl
,[31] Isoperimetric inequalities and capacities on Riemannian manifolds, Oper. Theory Adv. Appl. 109 (1999), 139 -153 | MR | Zbl
,[32] The Beurling operator for the hyperbolic plane, Ann. Acad. Sci. Fenn. Math. 37 (2012), 3 -18 | MR | Zbl
,[33] Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford (1993) | MR | Zbl
, , ,[34] Asymptotic Dirichlet problem for the p-Laplacian on Cartan–Hadamard manifolds, Proc. Amer. Math. Soc. 130 no. 11 (2002), 3393 -3400 | MR | Zbl
,[35] Non linear potential theory and quasiregular mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn. Ser. A 74 (1990) | MR | Zbl
,[36] p-Laplace operator, quasiregular mappings, and Picard-type theorems, Quasiconformal Mappings and Their Applications, Narosa, New Delhi (2007), 117 -150 | MR | Zbl
, ,[37] The role played by space dimension in elliptic critical problems, J. Differential Equations 156 (1999), 407 -426 | MR | Zbl
,[38] Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc. 361 no. 12 (2009), 6191 -6203 | MR | Zbl
, ,[39] Symmetric Green's functions on complete manifolds, Amer. J. Math. 109 (1987), 1129 -1154 | MR | Zbl
, ,[40] Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. Éc. Norm. Supér. (4) 39 no. 6 (2006), 921 -982 | EuDML | Numdam | MR | Zbl
, ,[41] On the best constant for Hardy's inequality in , Trans. Amer. Math. Soc. 350 (1998), 3237 -3255 | MR | Zbl
, , ,[42] The best possible constant in generalized Hardy's inequality for convex domain in , Nonlinear Anal. 28 (1997), 1601 -1610 | MR | Zbl
, ,[43] The sharp constant in Hardy's inequality for complement of bounded domain, Nonlinear Anal. 33 (1998), 105 -120 | MR | Zbl
, ,[44] Hardy's inequality for -functions on Riemannian manifolds, Proc. Amer. Math. Soc. 127 no. 9 (1999), 2745 -2754 | MR | Zbl
, ,[45] A simple approach to Hardy inequalities, Mat. Zametki 67 (2000), 563 -572 | MR | Zbl
,[46] A-priori estimates and blow-up of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math. 234 (2001), 1 -362 | Zbl
, ,[47] Vanishing and Finiteness Results in Geometric Analysis, Birkhäuser Verlag, Berlin (2008) | MR | Zbl
, , ,[48] Parabolicity of manifolds, Siberian Adv. Math. 9 (1999), 125 -150 | MR | Zbl
,[49] Solving the p-Laplacian on manifolds, Proc. Amer. Math. Soc. 128 no. 2 (2000), 541 -545 | MR | Zbl
,[50] The Gagliardo–Nirenberg inequalities and manifolds of non-negative Ricci curvature, J. Funct. Anal. 224 (2005), 230 -241 | MR | Zbl
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