Hardy, Sobolev, and CLR Inequalities

  • Alexander A. Balinsky
  • W. Desmond Evans
  • Roger T. Lewis
Part of the Universitext book series (UTX)


The Hardy and Sobolev inequalities are of fundamental importance in many branches of mathematical analysis and mathematical physics, and have been intensively studied since their discovery. A rich theory has been developed with the original inequalities on \((0,\infty )\) extended and refined in many ways, and an extensive literature on them now exists. We shall be focusing throughout the book on versions of the inequalities in L p spaces, with \(1 < p < \infty \). In this chapter we shall be mainly concerned with the inequalities in \((0,\infty )\) or \(\mathbb{R}^{n},\ n \geq 1\). Later in the chapter we shall also discuss the CLR (Cwikel, Lieb, Rosenbljum) inequality, which gives an upper bound to the number of negative eigenvalues of a lower semi-bounded Schrödinger operator in \(L^{2}(\mathbb{R}^{n})\). This has a natural place with the Hardy and Sobolev inequalities as the three inequalities are intimately related, as we shall show. Where proofs are omitted, e.g., of the Sobolev inequality, precise references are given, but in all cases we have striven to include enough background analysis to enable a reader to understand and appreciate the result.


Sobolev Inequality Extension Property Hardy Inequality Markov Generator Truncation Rule 
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.


  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Adimurthi, Tintarev, K.: Hardy inequalities for weighted Dirac operator. Ann. Mat. Pura Appl. 189, 241–251 (2010)Google Scholar
  3. 6.
    Allegretto, W.: Nonoscillation theory of elliptic equations of order 2n. Pac. J. Math. 64(1), 1–16 (1976)Google Scholar
  4. 9.
    Aubin, T.: Problemes isoperimetriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)zbMATHMathSciNetGoogle Scholar
  5. 15.
    Balinsky, A., Evans, W.D.: On the zero modes of Pauli operators. J. Funct. Anal. 179, 120–135 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 21.
    Balinsky, A., Evans, W.D., Umeda, T.: The Dirac-Hardy and Dirac-Sobolev inequalities in L 1. Publ. Res. Inst. Math. Sci. 47(3), 791–801 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 37.
    Chisholm, R.S., Everitt, W.N.: On bounded integral operators in the space of square integrable functions. Proc. R. Soc. Edinb. A 69, 199–204 (1971)zbMATHMathSciNetGoogle Scholar
  8. 38.
    Conlon, J.P.: A new proof of the Cwikel-Lieb-Rosenbljum bound. Rocky Mt. J.Math. 117–122 (1985)Google Scholar
  9. 39.
    Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106, 93–100 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 45.
    Davies, E.B., Hinz, A.M.: Explicit constants for Rellich inequalities in \(L_{p}(\Omega )\). Math. Z. 227(3), 511–523 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 46.
    Dolbeault, J., Esteban, M.J., Séré, E.: On the eigenvalues of operators with gaps. Application to Dirac operators. J. Funct. Anal. 174(1), 208–226 (2000)zbMATHGoogle Scholar
  12. 47.
    Dolbeault, J., Esteban, M.J., Loss, M., Vega, L.: An analytic proof of Hardy-like inequalities related to the Dirac operator. J. Funct. Anal. 216, 1–21 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 48.
    Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford University Press, Oxford (1987) [OX2 GDP]zbMATHGoogle Scholar
  14. 49.
    Edmunds, D.E., Evans, W.D.: Hardy Operators, Function Spaces, and Embeddings. Springer Monographs in Mathematics. Springer, Berlin/Heidelberg/New York (2004)zbMATHCrossRefGoogle Scholar
  15. 51.
    Esteban, M.J., Loss, M.: Self-adjointness of Dirac operators via Hardy-Dirac inequalities. J. Math. Phys. 48, 112107 (2007)MathSciNetCrossRefGoogle Scholar
  16. 59.
    Federer, H., Fleming, W.: Normal and integral currents. Ann. Math. 72, 458–520 (1960)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 67.
    Gagliardo, E.: Proprietaàdi alcune classi di funzionidi piu variabili. Ricerche Mat. 7, 102–137 (1958)zbMATHMathSciNetGoogle Scholar
  18. 73.
    Hardy, G.H.: An inequality between integrals. Messenger Math. 54, 150–156 (1925)Google Scholar
  19. 75.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)zbMATHGoogle Scholar
  20. 77.
    Herbst, I.: Spectral theory of the operator \((p^{2} + m^{2})^{1/2} - ze^{2}/r\). Commun. Math. Phys. 53(3), 285–294 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 83.
    Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin/Heidelberg/New York (1976)zbMATHCrossRefGoogle Scholar
  22. 91.
    Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy Inequality – About Its History and Some Related Results. Vydavatelský servis, Pilsen (2007)zbMATHGoogle Scholar
  23. 94.
    Landau, E.: A note on a theorem concerning series of positive terms. J. Lond. Math. Soc. 1, 38–39 (1926)zbMATHGoogle Scholar
  24. 103.
    Levin, D., Solomyak, M.: The Rozenbljum-Lieb-Cwikel inequality for Markov generators. J. Anal. Math. 71, 173–193 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 105.
    Lewis, R.T.: Singular elliptic operators of second order with purely discrete spectra. Trans. Am. Math. Soc. 271, 653–666 (1982)zbMATHCrossRefGoogle Scholar
  26. 109.
    Li, P., Yau, S.-T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88, 309–318 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 110.
    Lieb, E.H.: Bounds on the number of eigenvalues of Laplace and Schrödinger operators. Bull. Am. Math. Soc. 82, 751–753 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 111.
    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)Google Scholar
  29. 112.
    Lieb, E.H., Seiringer, R.: The Stability of Matter in Quantum Mechanics. Cambridge University Press, New York (2010)Google Scholar
  30. 118.
    Maz’ya, V.G.: Classes of domains and embedding theorems for function spaces. Dokl. Akad. Nauk. SSSR 133, 527–530 (1960). English transl.: Sov. Math. Dokl. 1, 882–885Google Scholar
  31. 121.
    Muckenhoupt, B.: Hardy’s inequality with weights. Stud. Math. 44, 31–38 (1972)zbMATHMathSciNetGoogle Scholar
  32. 124.
    Nirenberg, L.: On elliptic partial differential equations. Ann. Sc. Norm. Pisa 13, 1–48 (1959)MathSciNetGoogle Scholar
  33. 126.
    Opic, B., Kufner, A.: Hardy-type Inequalities. Pitman Research Notes in Mathematics, Series, vol. 219. Longman Science & Technology, Harlow (1990)Google Scholar
  34. 131.
    Rosenbljum, G.V.: The distribution of the discrete spectrum for singular differential operators. Soviet Math. Dokl. 13, 245–249 (1972)Google Scholar
  35. 134.
    Seiringer, R.: Inequalities for Schrödinger operators and applications to the stability of matter problem. Lectures given in Tucson, Arizona, 16–20 March 2009Google Scholar
  36. 137.
    Sobolev, S.L.: On a theorem of functional analysis. Mat. Sb. Am. Math. Soc. Transl. II Ser. 34, 39–68 (1938); 46, 471–497 (1963)Google Scholar
  37. 138.
    Solomyak, M.Z.: A remark on the Hardy inequalities. Integr. Equ. Oper. Theory 19, 120–124 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 140.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 147.
    Weder, R.A.: Spectral properties of one-body relativistic spin-zero Hamiltonians. Ann. Inst. H. Poincaré Sect. A (NS) 20, 211–220 (1974)MathSciNetGoogle Scholar
  40. 148.
    Weder, R.A.: Spectral analysis of pseudodifferential operators. J. Funct. Anal. 20(4), 319–337 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  41. 150.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. The University Press, Cambridge (1940)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alexander A. Balinsky
    • 1
  • W. Desmond Evans
    • 1
  • Roger T. Lewis
    • 2
  1. 1.School of MathematicsCardiff UniversityCardiffUK
  2. 2.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA

Personalised recommendations