Geodesic
Accuracy of constants in logarithmic Hardy-type inequalities in open multidimensional domains
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 155 (2013) no. 3, pp. 111-125 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Multidimensional Hardy-type inequalities in arbitrary domains of the Euclidean space are investigated. Accuracy of the constants is established in the case when the inner radius of the domain is finite, and the weight function contains degrees and the multiple logarithms of the function of the distance to the domain's boundary.
Keywords: Hardy-type inequalities, logarithmic singularity, accuracy of constants, distance function. 
@article{UZKU_2013_155_3_a11,
     author = {R. G. Nasibullin},
     title = {Accuracy of constants in logarithmic {Hardy-type} inequalities in open multidimensional domains},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {111--125},
     year = {2013},
     volume = {155},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2013_155_3_a11/}
}
TY  - JOUR
AU  - R. G. Nasibullin
TI  - Accuracy of constants in logarithmic Hardy-type inequalities in open multidimensional domains
JO  - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
PY  - 2013
SP  - 111
EP  - 125
VL  - 155
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/UZKU_2013_155_3_a11/
LA  - ru
ID  - UZKU_2013_155_3_a11
ER  - 
%0 Journal Article
%A R. G. Nasibullin
%T Accuracy of constants in logarithmic Hardy-type inequalities in open multidimensional domains
%J Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
%D 2013
%P 111-125
%V 155
%N 3
%U http://geodesic.mathdoc.fr/item/UZKU_2013_155_3_a11/
%G ru
%F UZKU_2013_155_3_a11
R. G. Nasibullin. Accuracy of constants in logarithmic Hardy-type inequalities in open multidimensional domains. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 155 (2013) no. 3, pp. 111-125. http://geodesic.mathdoc.fr/item/UZKU_2013_155_3_a11/

[1] Dubinskii Yu. A., “Ob odnom neravenstve tipa Khardi i ego prilozheniyakh”, Trudy MIAN, 269, 2010, 112–132 | MR | Zbl

[2] Davies E. B., “A review of Hardy Inequalities”, Oper. Theory Adv. Appl., 110 (1999), 55–67 | MR | Zbl

[3] Laptev A., Weidl T., “Hardy inequalities for magnetic Dirichlet forms”, Oper. Theory Adv. Appl., 108 (1999), 299–305 | MR

[4] Balinsky A., Laptev A., Sobolev A. V., “Generalized Hardy inequality for the magnetic Dirichlet forms”, J. Stat. Phys., 116:1–4 (2004), 507–521 | DOI | MR | Zbl

[5] Hardy G. H., Littlewood J. E., Pólya G., Inequalities, Cambridge Univ. Press, Cambridge, 1988, 340 pp. | MR | Zbl

[6] Opic B., Kufner A., Hardy-type Inequalities, Longman Scientific Technical, Essex, England, 1990, 333 pp. | MR | Zbl

[7] Shum D. T., “On integral inequalities related to Hardy's”, Can. Math. Bull., 14:2 (1971), 225–230 | DOI | MR | Zbl

[8] Chan L.-Y., “Some extensions of Hardy's inequality”, Can. Math. Bull., 22:2 (1979), 165–169 | DOI | MR | Zbl

[9] Pachpatte B. G., “A note on certain inequalities related to Hardy's inequality”, Indian J. Pure Appl. Math., 23:11 (1992), 773–776 | MR | Zbl

[10] Solomyak M., “A remark on the Hardy inequalities”, Integr. Equat. Oper. Th., 19:1 (1994), 120–124 | DOI | MR | Zbl

[11] Pec̆arić J. E., Love E. R., “Still more generalization of Hardy's inequality”, J. Austral. Math. Soc. Series A, 59 (1995), 214–224 | DOI | MR

[12] Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Laptev A., “A geometrical version of Hardy's inequality”, J. Funct. Anal., 189:2 (2002), 539–548 | DOI | MR | Zbl

[13] Tidblom J., “A geometrical version of Hardy's inequality for $W^{1,p}_0(\Omega)$”, Proc. Amer. Math. Soc., 132:8 (2004), 2265–2271 | DOI | MR | Zbl

[14] Avkhadiev F. G., “Neravenstva tipa Khardi v ploskikh i prostranstvennykh otkrytykh mnozhestvakh”, Trudy MIAN, 255, 2006, 8–18 | MR

[15] del Pino M., Dolbeault J., Filippas S., Tertikas A., “A logarithmic Hardy inequality”, J. Funct. Anal., 259:8 (2010), 2045–2072 | DOI | MR | Zbl

[16] Avkhadiev F. G., Nasibullin R. G., Shafigullin I. K., “Neravenstva tipa Khardi so stepennymi i logarifmicheskimi vesami v oblastyakh evklidova prostranstva”, Izv. vuzov. Matem., 2011, no. 9, 90–94 | MR | Zbl

[17] Avkhadiev F. G., “Hardy type inequalities in higher dimensions with explicit estimate of constants”, Lobachevskii J. Math., 21 (2006), 3–31 | MR | Zbl

[18] Avkhadiev F. G., Wirths K.-J., “Unified Poincaré and Hardy inequalities with sharp constants for convex domains”, Z. Angew. Math. Mech., 87:8–9 (2007), 632–642 | DOI | MR | Zbl

[19] Avkhadiev F. G., Wirths K.-J., “Weighted Hardy inequalities with sharp constants”, Lobachevskii J. Math., 31:1 (2010), 1–7 | DOI | MR | Zbl

[20] Avkhadiev F. G., Wirths K.-J., “Sharp Hardy-type inequalities with Lamb's constants”, Bull. Belg. Math. Soc. Simon Stevin, 18:11 (2011), 723–736 | MR | Zbl

[21] Mikhlin S. G., Lineinye uravneniya v chastnykh proizvodnykh, Vyssh. shk., M., 1977, 431 pp. | MR

[22] Sobolev L. S., Izbrannye voprosy teorii funktsionalnykh prostranstv i obobschennykh proizvodnykh, Nauka, M., 1989, 254 pp. | MR

[23] Natanson I. P., Konstruktivnaya teoriya funktsii, Gos. izd-vo tekhn.-teoret. lit., M., 1949, 688 pp. | MR