Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities (Cambridge University Press, Cambridge, 1973).
V. G. Maz’ya, Sobolev Spaces (Springer-Verlag, Berlin, New York, 1985).
A. Ancona, “On Strong Barriers and an Inequality of Hardy for Domains in R n,” J. London Math. Soc. (2) 37, 274–290 (1986).
V. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Research Notes in Math. (Longman, Harlowe, 1990), V. 219.
H. Brezis and M. Marcus, “Hardy’s Inequalities Revisited,” Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 25(1–2), 217–237 (1997).
F. G. Avkhadiev, “Solution of the Generalized Saint Venant Problem,” Matem. Sborn. 189(12), 1739–1748 (1998).
M. Marcus, V. J. Mizel, and Y. Pinchover, “On the Best Constants for Hardy’s Inequality in R n,” Trans. Amer. Math. Soc. 350(8), 3237–3250 (1998).
V. M. Miklyukov and M. K. Vuorinen, “Hardy’s Inequalities forW 1,p0 -Functions on Riemannian Manifolds,” Proc. Amer. Math. Soc. 127(9), 2145–2154 (1999).
E. B. Davies, “A Review of Hardy Inequalities,” The Maz’ya anniversary collection. Vol. 2. Oper. Theory Adv. Appl. 110, 55–67 (1999).
M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and A. Laptev, “A Geometrical Version of Hardy’s Inequality,” J. Funct. Anal. 189(2), 539–548 (2002).
A. Balinsky, A. Laptev, and A. V. Sobolev, “Generalized Hardy Inequality for the Magnetic Dirichlet Forms,” J. Statistical Physics 116(1–4), 507–521 (2004).
F. G. Avkhadiev, “Conformally Invariant Inequalities in Mathematical Physics,” Naukoemkie tekhnologii 5(4), 47–51 (2004).
Ch. Pommerenke, “Uniformly Perfect Sets and the Poincaré Metric,” Arch. Math. 32, 192–199 (1979).
J. B. Garnett and D. E. Marschall, Harmonic Measure (Cambridge University Press, Cambridge, 2005).
F. G. Avkhadiev, “Hardy Type Inequalities in Higher Dimensions with Explicit Estimate of Constants,” Lobachevskii J. Math. XXI, 3–31 (2006) (http://ljm.ksu.ru).
S. Filippas, V. G. Maz’ya, and A. Tertikas, “On a Question of Brezis and Marcus,” Calc. Var. Partial Differential Equations 25(4), 491–501 (2006).
F. G. Avkhadiev, “Hardy-Type Inequalities on Planar and Spatial Open Sets,” Trudy V. A. Steklov Matem. Inst. Ross. Akad. Nauk 255, 2–12 (2006).
F. G. Avkhadiev and K.-J. Wirths, “Unified Poincaréand Hardy Inequalities with Sharp Constants for Convex Domains,” Z. Angew.Math. Mech. 87(8–9), 632–642 (2007).
R. D. Benguria, R. L. Franc, and M. Loss, “The Sharp Constants in the Hardy-Sobolev-Maz’ya Inequality in the Three Dimensional Upper Half-Space,” Math. Res. Lett. 15(4), 613–622 (2008).
R. L. Franc and R. Seiringer, “Non-Linear Ground State Representation and Sharp Hardy Inequalities,” J. Funct. Anal. 255(12), 3407–3430 (2008).
F. G. Avkhadiev and K.-J. Wirths, Schwarz-Pick Type Inequalities (Birkhäuser, Boston-Berlin-Bern, 2009).
F. G. Avkhadiev and K.-J. Wirths, “Weighted Hardy Inequalities with Sharp Constants,” Lobachevskii J. Math. 31(1), 1–7 (2010).
F. Avkhadiev and A. Laptev, “Hardy Inequalities for Nonconvex Domains,” in Intern. Math. Series “Around Research of Vladimir Maz’ya. I.” Function Spaces, Ed. by A. Laptev 11, 1–12 (2010).
M. Del Pino, J. Dolbeault, S. Filippas, and A. Tertikas, “A Logarithmic Hardy Inequality,” J. Funct. Anal. 259(8), 2045–2072 (2010).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © F.G. Avkhadiev, R.G. Nasibullin, and I.K. Shafigullin, 2011, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2011, No. 9, pp. 90–94.
About this article
Cite this article
Avkhadiev, F.G., Nasibullin, R.G. & Shafigullin, I.K. Hardy-type inequalities with power and logarithmic weights in domains of the Euclidean space. Russ Math. 55, 76–79 (2011). https://doi.org/10.3103/S1066369X1109009X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X1109009X