Hardy inequalities with non-standard remainder terms

Cianchi, Andrea ; Ferone, Adele

Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008), p. 889-906 / Harvested from Numdam

Détail

Publié le : 2008-01-01

MR 2457816 | Zbl 1153.26310

DOI : https://doi.org/10.1016/j.anihpc.2007.05.003

Bibliographie

[1] Adams R.A., Sobolev Spaces, Academic Press, Orlando, 1975. | MR 450957 | Zbl 0314.46030

[2] Adimurthi , Chaudhuri N., Ramaswamy M., An improved Hardy-Sobolev inequality and its applications, Proc. Amer. Math. Soc. 130 (2002) 489-505. | MR 1862130 | Zbl 0987.35049

[3] Adimurthi , Esteban M.J., An improved Hardy-Sobolev inequality in ${W}^{1,p}$ and its applications to Schrödinger operators, NoDEA Nonlin. Differential Equation Appl. 12 (2005) 243-263. | MR 2184082 | Zbl 1089.35035

[4] Barbatis G., Filippas S., Tertikas A., A unified approach to improved ${L}^{p}$ Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004) 2169-2196. | MR 2048514 | Zbl 1129.26019

[5] Bartsch T., Weth T., Willem M., A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calc. Var. Part. Diff. Eq. 18 (2003) 57-75. | MR 2018667 | Zbl 1059.31006

[6] Bennett C., Sharpley R., Interpolation of Operators, Academic Press, Boston, 1988. | MR 928802 | Zbl 0647.46057

[7] Bianchi G., Egnell H., A note on the Sobolev inequality, J. Funct. Anal. 100 (1991) 18-24. | MR 1124290 | Zbl 0755.46014

[8] Brezis H., Lieb E., Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985) 73-86. | MR 790771 | Zbl 0577.46031

[9] Brezis H., Marcus M., Shafrir I., Extremal functions for Hardy's inequality with weights, J. Funct. Anal. 171 (2000) 177-191. | MR 1742864 | Zbl 0953.26006

[10] Brezis H., Nirenberg L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983) 437-477. | MR 709644 | Zbl 0541.35029

[11] Brezis H., Wainger S., A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Part. Diff. Eq. 5 (1980) 773-789. | MR 579997 | Zbl 0437.35071

[12] Brothers J.E., Ziemer W.P., Minimal rearrangements of Sobolev functions, J. Reine Angew. Math. 384 (1988) 419-431. | MR 929981 | Zbl 0633.46030

[13] Cianchi A., A quantitative Sobolev inequality in BV, J. Funct. Anal. 237 (2006) 466-481. | MR 2230346 | Zbl 1110.46020

[14] A. Cianchi, Sharp Morrey-Sobolev inequalities and the distance from extremals, Trans. Amer. Math. Soc., in press. | MR 2395175 | Zbl 1153.46019

[15] A. Cianchi, A. Ferone, A strenghtened version of the Hardy-Littlewood inequality, preprint. | Zbl 1155.26017

[16] A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, The sharp Sobolev inequality in quantitative form, preprint. | MR 2538501 | Zbl pre05608955

[17] Cwikel M., Pustylnik E., Sobolev type embeddings in the limiting case, J. Fourier Anal. Appl. 4 (1998) 433-446. | MR 1658620 | Zbl 0930.46027

[18] Detalla A., Horiuchi T., Ando H., Missing terms in Hardy-Sobolev inequalities and its applications, Far East J. Math. Sci. 14 (2004) 333-359. | MR 2108051 | Zbl 1109.26019

[19] Dolbeault J., Esteban M.J., Loss M., Vega L., An analytical proof of Hardy-like inequalities related to the Dirac operator, J. Funct. Anal. 216 (2004) 1-21. | MR 2091354 | Zbl 1060.35120

[20] Edmunds D.E., Kerman R.A., Pick L., Optimal Sobolev imbeddings involving rearrangement invariant quasi-norms, J. Funct. Anal. 170 (2000) 307-355. | MR 1740655 | Zbl 0955.46019

[21] N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative Sobolev inequality in BV, preprint.

[22] Filippas S., Maz'Ya V.G., Tertikas A., On a question of Brezis and Marcus, Calc. Var. Part. Diff. Eq. 25 (2006) 491-501. | MR 2214621 | Zbl 1121.26014

[23] Gazzola F., Grunau H.C., Mitidieri E., Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc. 356 (2004) 2149-2168. | MR 2048513 | Zbl 1079.46021

[24] Gazzola F., Grunau H.C., Squassina M., Existence and non-existence results for critical growth biharmonic elliptic equations, Calc. Var. Part. Diff. Eq. 18 (2003) 117-143. | MR 2010961 | Zbl pre02004119

[25] Hansson K., Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979) 77-102. | MR 567435 | Zbl 0437.31009

[26] Kawohl B., Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., vol. 1150, Springer-Verlag, Berlin, 1985. | MR 810619 | Zbl 0593.35002

[27] Maz'Ya V.G., Sobolev Spaces, Springer-Verlag, Berlin, 1985. | MR 817985 | Zbl 0692.46023

[28] Opic B., Kufner A., Hardy-type Inequalities, Longman Scientific Technical, Harlow, 1990. | MR 1069756 | Zbl 0698.26007

[29] Pohozhaev S.I., On the imbedding Sobolev theorem for $pl=n$, in: Doklady Conference, Section Math., Moscow Power Inst., 1965, pp. 158-170, (Russian).

[30] Talenti G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976) 353-372. | MR 463908 | Zbl 0353.46018

[31] Tidblom J., A Hardy inequality in the half-space, J. Funct. Anal. 221 (2005) 482-495. | MR 2124873 | Zbl 1077.26010

[32] Trudinger N.S., On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473-483. | MR 216286 | Zbl 0163.36402

[33] Vazquez J.L., Zuazua E., The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal. 173 (2000) 103-153. | MR 1760280 | Zbl 0953.35053

[34] Yudovich V.I., Some estimates connected with integral operators and with solutions of elliptic equations, Soviet Math. Dokl. 2 (1961) 746-749. | Zbl 0144.14501