Hardy-type inequalities with power and logarithmic weights in domains of the Euclidean space
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2011), pp. 90-94.

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

We consider Hardy-type inequalities in domains of the Euclidean space for the case, when the weight depends on the distance function to the domain boundary and has power and logarithmic singularities. We prove several new inequalities with sharp constants.
Keywords: Hardy-type inequalities, distance function to the boundary, iterations of logarithms.
@article{IVM_2011_9_a8,
     author = {F. G. Avkhadiev and R. G. Nasibullin and I. K. Shafigullin},
     title = {Hardy-type inequalities with power and logarithmic weights in domains of the {Euclidean} space},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {90--94},
     publisher = {mathdoc},
     number = {9},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2011_9_a8/}
}
TY  - JOUR
AU  - F. G. Avkhadiev
AU  - R. G. Nasibullin
AU  - I. K. Shafigullin
TI  - Hardy-type inequalities with power and logarithmic weights in domains of the Euclidean space
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2011
SP  - 90
EP  - 94
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2011_9_a8/
LA  - ru
ID  - IVM_2011_9_a8
ER  - 
%0 Journal Article
%A F. G. Avkhadiev
%A R. G. Nasibullin
%A I. K. Shafigullin
%T Hardy-type inequalities with power and logarithmic weights in domains of the Euclidean space
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2011
%P 90-94
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2011_9_a8/
%G ru
%F IVM_2011_9_a8
F. G. Avkhadiev; R. G. Nasibullin; I. K. Shafigullin. Hardy-type inequalities with power and logarithmic weights in domains of the Euclidean space. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2011), pp. 90-94. http://geodesic.mathdoc.fr/item/IVM_2011_9_a8/

[1] Hardy G. H., Littlewood J. E., Polya G., Inequalities, Cambridge University Press, Cambridge, 1973

[2] Maz'ya V. G., Sobolev spaces, Springer-Verlag, Berlin–New York, 1985

[3] Ancona A., “On strong barriers and an inequality of Hardy for domains in $R^n$”, J. London Math. Soc. (2), 37 (1986), 274–290 | DOI | MR

[4] Opic V., Kufner A., Hardy-type inequalities, Pitman Research Notes in Math., 219, Longman, Harlow, 1990 | MR | Zbl

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

[6] Avkhadiev F. G., “Reshenie obobschennoi zadachi Sen-Venana”, Matem. sb., 189:12 (1998), 3–12 | MR | Zbl

[7] Marcus M., Mizel V. J., Pinchover Y., “On the best constants for Hardy's inequality in $R^n$”, Trans. Amer. Math. Soc., 350:8 (1998), 3237–3250 | DOI | MR

[8] Miklyukov V. M., Vuorinen M. K., “Hardy's inequalities for $W_0^{1,p}$-functions on Riemannian manifolds”, Proc. Amer. Math. Soc., 127:9 (1999), 2145–2154 | DOI | MR

[9] Davies E. B., “A review of Hardy inequalities”, The Maz'ya anniversary collection, v. 2, Oper. Theory Adv. Appl., 110, 1999, 55–67 | Zbl

[10] 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

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

[12] Avkhadiev F. G., “Konformno-invariantnye neravenstva matematicheskoi fiziki”, Naukoemkie tekhnologii, 5:4 (2004), 47–51

[13] Pommerenke Ch., “Uniformly perfect sets and the Poincaré metric”, Arch. Math., 32 (1979), 192–199 | DOI | MR | Zbl

[14] Garnett J. B., Marschall D. E., Harmonic measure, Cambridge University Press, Cambridge, 2005 | Zbl

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

[16] Filippas S., Maz'ya V. G., Tertikas A., “On a question of Brezis and Marcus”, Calc. Var. Partial Differential Equations, 25:4 (2006), 491–501 | DOI | MR | Zbl

[17] Avkhadiev F. G., “Neravenstva tipa Khardi v ploskikh i prostranstvennykh otkrytykh mnozhestvakh”, Tr. matem. in-ta RAN im. V. A. Steklova, 255, 2006, 8–18 | MR

[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] Benguria R. D., Franc R. L., Loss M., “The sharp constants in the Hardy–Sobolev–Maz'ya inequality in the three dimensional upper half-space”, Math. Res. Lett., 15:4 (2008), 613–622 | MR | Zbl

[20] Franc R. L., Seiringer R., “Non-linear ground state representation and sharp Hardy inequalities”, J. Funct. Anal., 255:12 (2008), 3407–3430 | DOI | MR

[21] Avkhadiev F. G., Wirths K.-J., Schwarz–Pick type inequalities, Birkhäuser, Boston–Berlin–Bern, 2009 | Zbl

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

[23] Avkhadiev F., Laptev A., “Hardy inequalities for nonconvex domains”, Around Research of Vladimir Maz'ya, v. I, Intern. Math. Series, 11, Function Spaces, ed. Laptev A., 2010, 1–12 | MR | Zbl

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