Geodesic
Hardy type inequalities with logarithmic and power weights for a special family of non-convex domains
Ufa mathematical journal, Tome 5 (2013) no. 2, pp. 43-55

Voir la notice de l'article provenant de la source Math-Net.Ru

In the present work we obtain variational Hardy type inequalities with power and logarithmic weights which are generalizations of the corresponding inequalities given earlier in the papers by M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, and J. Tidblom. We formulate and prove inequalities for arbitrary domains, and then we substantially simplify them for the class of convex domains and a special family of non-convex domains.
Keywords: Hardy type inequalities, regular domains, distance function, iteration of logarithms.
Mots-clés : convex domains 
R. G. Nasibullin; A. M. Tukhvatullina. Hardy type inequalities with logarithmic and power weights for a special family of non-convex domains. Ufa mathematical journal, Tome 5 (2013) no. 2, pp. 43-55. http://geodesic.mathdoc.fr/item/UFA_2013_5_2_a4/
@article{UFA_2013_5_2_a4,
     author = {R. G. Nasibullin and A. M. Tukhvatullina},
     title = {Hardy type inequalities with logarithmic and power weights for a special family of non-convex domains},
     journal = {Ufa mathematical journal},
     pages = {43--55},
     year = {2013},
     volume = {5},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2013_5_2_a4/}
}
TY  - JOUR
AU  - R. G. Nasibullin
AU  - A. M. Tukhvatullina
TI  - Hardy type inequalities with logarithmic and power weights for a special family of non-convex domains
JO  - Ufa mathematical journal
PY  - 2013
SP  - 43
EP  - 55
VL  - 5
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/UFA_2013_5_2_a4/
LA  - en
ID  - UFA_2013_5_2_a4
ER  - 
%0 Journal Article
%A R. G. Nasibullin
%A A. M. Tukhvatullina
%T Hardy type inequalities with logarithmic and power weights for a special family of non-convex domains
%J Ufa mathematical journal
%D 2013
%P 43-55
%V 5
%N 2
%U http://geodesic.mathdoc.fr/item/UFA_2013_5_2_a4/
%G en
%F UFA_2013_5_2_a4

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

[2] A. Balinsky, A. Laptev, A. V. Sobolev, “Generalized Hardy inequality for the magnetic Dirichlet forms”, Journal of statistical physics, 116 (2004), 507–521 | DOI | MR | Zbl

[3] A. Laptev, T. Weidl, “Hardy inequalities for magnetic Dirichlet forms”, Operator Theory: Advances and Applications, 108, 1999, 299–305 | MR

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

[5] E. B. Davies, Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, 42, Cambridge Univ. Press, Cambridge, 1995, 186 pp. | MR | Zbl

[6] E.B. Davies, “A Review of Hardy Inequalities”, The Maz'ya anniversary Collection, v. 2, Oper. Theory Adv. Appl., 110, 1999, 55–67 | MR | Zbl

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

[8] Avkhadiev F. G., Neravenstva dlya integralnykh kharakteristik oblastei, KGU, Kazan, 2006, 140 pp.

[9] F. G. Avkhadiev, “Hardy type inequalities in higher dimensions with explicit estimate of constants”, Lobachevskii J. Math., XXI (2006), 3–31 http://ljm.ksu.ru | MR | Zbl

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

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

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

[13] H. Brezis, M. Marcus, “Hardy's inequality revisited”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25:1–2 (1997), 217–237 | MR | Zbl

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

[15] Tidblom J., “A geometrical version of Hardy's inequality for $ W^{1,p}_0(\Omega)$”, Proc. Amer. Math. Soc., 2004, no. 132, 2265–2271 | DOI | MR | Zbl

[16] G. Barbatis, S. Filippas, A. Tertikas, “Refined geometric $L^p$ Hardy inequalities”, Communications in Contemporary Mathematics, 5:6 (2003), 869–881 | DOI | MR | Zbl

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

[18] S. M. Buckley, R. Hurri-Syrjänene, Iterated log-scale Orlicz–Hardy inequalities, Preprint, Department of Mathematics, National University of Ireland, Maynooth, 2011 | MR

[19] Nasibullin R. G., “Neravenstva tipa Khardi, vklyuchayuschie povtornye logarifmy”, Trudy matematicheskogo tsentra im. N. I. Lobachevskogo, 43, Izd-vo Kazanskogo matematicheskogo obschestva, Kazan, 2011, 262–263

[20] Nasibullin R. G., “Nekotoroe obobschenie neravenstva Khardi”, Trudy matematicheskogo tsentra im. N. I. Lobachevskogo, 44, Izd-vo Kazanskogo matematicheskogo obschestva, Kazan, 2011, 221–222

[21] Tukhvatullina A. M., “Neravenstva tipa Khardi dlya spetsialnogo semeistva nevypuklykh oblastei”, Uchen. zap. Kazan. gos. un-ta. Ser. Fiz.-matem. nauki, 153:1 (2011), 211–220 | Zbl

[22] Tukhvatullina A. M., “Rasprostranenie kriteriya regulyarnosti oblasti Devisa na mnogomernye oblasti i ego primenenie v neravenstvakh tipa Khardi”, Trudy matematicheskogo tsentra im. N. I. Lobachevskogo, 43, Izd-vo Kazanskogo matematicheskogo obschestva, Kazan, 2011, 350–351

[23] Tukhvatullina A. M., “Dostatochnoe uslovie regulyarnosti oblasti i ego primenenie v neravenstvakh tipa Khardi”, Trudy matematicheskogo tsentra im. N. I. Lobachevskogo, 38, Izd-vo Kazanskogo matematicheskogo obschestva, Kazan, 2009, 285–287