A Collection of Sharp Dilation Invariant Integral Inequalities for Differentiable Functions

  • Vladimir Maz'ya
  • Tatyana Shaposhnikova
Part of the International Mathematical Series book series (IMAT, volume 8)

We find best constants in several dilation invariant integral inequalities involving derivatives of functions. Some of these inequalities are new and some were known without best constants. In particular, we deal with an estimate for a quadratic form of the gradient, weighted Gårding inequality for the biharmonic operator, dilation invariant Hardy's inequalities with remainder term, a generalized Hardy–Sobolev inequality with sharp constant, and the Hardy inequality with sharp Sobolev remainder term.


Sobolev Inequality Remainder Term Hardy Inequality Good Constant Sharp Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adimurthi: Hardy–Sobolev inequalities inH(Ω) and its applications. Commun. Con-temp. Math.4, no. 3, 409–434 (2002)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Adimurthi, Chaudhuri, N., Ramaswamy, M.: An improved Hardy–Sobolev inequality and its applications. Proc. Am. Math. Soc.130no. 489–505 (2002)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Adimurthi, Grossi, M., Santra, S.: Optimal Hardy—Rellich inequalities, maximum principle and related eigenvalue problems. J. Funct. Anal.240, no.1, 36–83 (2006)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Aubin, T.: Problèmes isopérimètriques et espaces de Sobolev. C.R. Acad. Sci., Paris280, no. 5, 279–281 (1975)MATHMathSciNetGoogle Scholar
  5. 5.
    Barbatis, G., Filippas, S., Tertikas, A.: Series expansion forL pHardy inequalities. Indiana Univ. Math. J.52, no. 1, 171–190. (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Barbatis, G., Filippas, S., Tertikas, A.: A unified approach to improvedL pHardy inequalities with best constants. Trans. Am. Math. Soc.356, no. 6, 2169–2196. (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Benguria, R.D., Frank, R.L., Loss, M.: The sharp constant in the Hardy–Sobolev— Maz'ya inequality in the three dimensional upper half-space. arXiv: math/0705 3833Google Scholar
  8. 8.
    Brandolini, B., Chiacchio, F., Trombetti, C.: Hardy inequality and Gaussian measure. Commun. Pure Appl. Math.6, no. 2, 411–428, (2007)MATHMathSciNetGoogle Scholar
  9. 9.
    Brezis, H., Marcus, M.: Hardy's inequalities revisited. Dedicated to Ennio De Giorgi. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4)25. no. 1–2, 217–237 (1997)MATHMathSciNetGoogle Scholar
  10. 10.
    Brezis, H., Vázquez, J.L.: Blow-up solutions of some noinlinear elliptic problems. Rev. Mat. Univ. Comp. Madrid10, 443–469 (1997)MATHGoogle Scholar
  11. 11.
    Catrina, F., Wang, Z.-Q.: On the Caffarelli—Kohn—Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Commun. Pure Appl. Math.54, 229–258, (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Costin, O., Maz'ya, V.: Sharp Hardy—Leray inequality for axisymmetric divergence-free fields. Calc. Var. Partial Differ. Equ. [To appear]; arXiv:math/0703116Google Scholar
  13. 13.
    Davies, E.B.: A review of Hardy inequalities. In: The Maz'ya Anniversary Collection. Vol. 2 (Rostock, 1998), pp. 55–67, Oper. Theory Adv. Appl. 110, Birkhäuser, Basel (1999)Google Scholar
  14. 14.
    Dávila, J., Dupaigne, L.: Hardy type inequalities. J. Eur. Math. Soc.6, no. 3, 335–365 (2004)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Dou, J., Niu, P., Yuan, Z.: A Hardy inequality with remainder terms in the Heisenberg group and weighted eigenvalue problem. J. Inequal. Appl.2007, ID 32585 (2007)Google Scholar
  16. 16.
    Eilertsen, S.: On weighted positivity and the Wiener regularity of a boundary point for the fractional Laplacian. Ark. Mat.38, 53–75 (2000)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Evans, W.E., Lewis, R.T.: Hardy and Rellich inequalities with remainders. J. Math. Ineq. 1, no. 4, 473–490 (2007)MATHMathSciNetGoogle Scholar
  18. 18.
    Federer, H., Fleming, W.H.: Normal and integral currents. Ann. Math.72, 458–520 (1960)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Filippas, S., Maz'ya, V., Tertikas, A.: Sharp Hardy–Sobolev inequalities. C.R. Acad. Sci., Paris339, no. 7, 483–486 (2004)MATHMathSciNetGoogle Scholar
  20. 20.
    Filippas, S., Maz'ya, V., Tertikas, A.: On a question of Brezis and Marcus. Calc. Var. Partial Differ. Equ.25, no. 4, 491–501 (2006)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Filippas, S., Maz'ya, V., Tertikas, A.: Critical Hardy–Sobolev inequalities J. Math. Pure Appl.87, 37–56 (2007)MATHMathSciNetGoogle Scholar
  22. 22.
    Filippas, S., Tertikas, A.: Optimizing improved Hardy inequalities. J. Funct. Anal.192, 186–233 (2002)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Filippas, S., Tertikas, A., Tidblom, J.: On the structure of Hardy–Sobolev—Maz'ya inequalities. arXiv: math/0802.0986Google Scholar
  24. 24.
    Frank, R.L., Seiringer, R.: Nonlinear ground State representations and sharp Hardy inequalities. arXiv: math/0803.0503Google Scholar
  25. 25.
    Gagliardo, E.; Ulteriori proprietà di alcune classi di funzioni in pi`u variabili. Ric. Mat.8, no. 1, 24–51 (1959)MATHMathSciNetGoogle Scholar
  26. 26.
    Glazer, V., Martin, A., Grosse, H., Thirring, W.: A family of optimal conditions for the absence of bound States in a potential. In: Studies in Mathematical Physics, pp. 169–194. Princeton Univ. Press, Princeton, NJ (1976)Google Scholar
  27. 27.
    Gazzola, F., Grunau, H.-Ch., Mitidieri, E.: Hardy inequalities with optimal constants and remainder terms. Trans. Am. Math. Soc.356, 2149–2168 (2004)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge Univ. Press, Cambridge (1952)MATHGoogle Scholar
  29. 29.
    Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Laptev, A.: A geometrical version of Hardy's inequality. J. Funct. Anal.189, 539–548 (2002)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Il'in, V.P.: Some integral inequalities and their applications in the theory of differen-tiable functions of several variables (Russian). Mat. Sb54, no. 3, 331–380 (1961)MathSciNetGoogle Scholar
  31. 31.
    Lieb, E.H., Sharp constants in the Hardy—Littlewood-Sobolev and related inequalities. Ann. Math.118, 349–374 (1983)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Mayboroda, S., Maz'ya, V.: Boundedness of the Hessian of a biharmonic function in a convex domain. Commun. Partial Differ. Equ. [To appear]; arXiv:math/0611058Google Scholar
  33. 33.
    Maz'ya, V.: Classes of domains and imbedding theorems for function spaces (Russian). Dokl. Akad. Nauk SSSR133, 527–530 (1960); English transl.: Sov. Math., Dokl. 1, 882–885 (1960)Google Scholar
  34. 34.
    Maz'ya, V.: On a degenerating problem with directional derivative (Russian). Mat. Sb.87, 417–454, (1972); English transl.: Math. USSR Sb.16, no. 3, 429–469 (1972)MathSciNetGoogle Scholar
  35. 35.
    Maz'ya, V.: On certain integral inequalities for functions of many variables (Russian). Probl. Mat. Anal.3, 33–68 (1972); English transl.: J. Math. Sci. 1, no. 2, 205–234 (1973)Google Scholar
  36. 36.
    Maz'ya, V.: Behavior of solutions to the Dirichlet problem for the biharmonic operator at a boundary point. Equadiff IV (Proc. Czechoslovak Conf. Differential Equations and their Applications, Prague, 1977) pp. 250–262. Lect. Notes Math.703. Springer, Berlin (1979)Google Scholar
  37. 37.
    Maz'ya, V.G.: Sobolev Spaces. Springer-Verlag, Berlin—Tokyo (1985)Google Scholar
  38. 38.
    Maz'ya, V.: On the Wiener-type regularity of a boundary point for the polyharmonic operator. Appl. Anal.71, 149–165 (1999)MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Maz'ya, V.: The Wiener test for higher order elliptic equations. Duke Math. J.115. no. 3, 479–512 (2002)MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Maz'ya, V.: Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces. Contemp. Math.338, 307–340 (2003)MathSciNetGoogle Scholar
  41. 41.
    Maz'ya, V.: Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolev type imbeddings. J. Funct. Anal..224, no. 2, 408– 430 (2005)MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Maz'ya, V.: Conductor inequalities and criteria for Sobolev type two-weight imbeddings. J. Comp. Appl. Math..194, no. 1. 94–114 (2006)MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Maz'ya, V., Tashchiyan, G.: On the behavior of the gradient of the solution of the Dirichlet problem for the biharmonic equation near a boundary point of a three-dimensional domain (Russian). Sib. Mat. Zh.31, no. 6, 113–126 (1991); English transl.: Sib. Math. J.31, no. 6, 970–983 (1991)MathSciNetGoogle Scholar
  44. 44.
    Mikhlin, S., Prössdorf, S.: Singular Integral Operators. Springer (1986)Google Scholar
  45. 45.
    Nazarov, A.I.: Hardy–Sobolev inequalities in a cone. J. Math. Sci.132, no. 4, 419–427 (2006) Hardy–Sobolev inequalities in a cone (Russian). Probl. Mat. Anal.31, 39–46 (2005); English transl.: J. Math. Sci., New York132, no. 4, 419–427 (2006)CrossRefMathSciNetGoogle Scholar
  46. 46.
    Nirenberg, L.: On elliptic partial differential equations, Lecture II. Ann. Sc. Norm. Super. Pisa, Ser. 313, 115–162 (1959)MathSciNetGoogle Scholar
  47. 47.
    Rosen, G.: Minimum value forCin the Sobolev inequality ǁφз ǁ ⩽Cǁgradφǁз, SIAM J. Appl. Math.21, no. 1, 30–33 (1971)MATHGoogle Scholar
  48. 48.
    Sobolev, S.L.: On a theorem of functional analysis (Russian). Mat. Sb.46, 471–497 (1938); English transl.: Am. Math. Soc, Transl., II. Ser.34, 39–68 (1963)Google Scholar
  49. 49.
    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, Princeton, NJ (1971)MATHGoogle Scholar
  50. 50.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl.136, 353–172 (1976)MathSciNetGoogle Scholar
  51. 51.
    Tertikas, A., Tintarev, K.: On existence of minimizers for the Hardy–Sobolev—Maz'ya inequality. Ann. Mat. Pura Appl.186, no. 4, 645–662 (2007)MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Tertikas, A., Zographopoulos, N.B.: Best constants in the Hardy-Rellich inequalities and related improvements. Adv. Math.209, no. 2, 407–459 (2007)MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Tidblom, J.: A geometrical version of Hardy's inequality forW 0 1,p. Proc. Am. Math. Soc.138, no. 8, 2265–2271 (2004)CrossRefMathSciNetGoogle Scholar
  54. 54.
    Tidblom, J.: A Hardy inequality in the half-space. J. Funct. Anal.221, 482–495 (2005)MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Vázquez, J.L., Zuazua, E.: The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential. J. Funct. Anal.173, 103–153 (2000)MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Yaotian, S., Zhihui, C: General Hardy inequalities with optimal constants and remainder terms. J. Inequal. Appl. no. 3, 207–219 (2005)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Vladimir Maz'ya
    • Tatyana Shaposhnikova

      There are no affiliations available

      Personalised recommendations