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A Survey of Some Norm Inequalities

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Abstract

We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type

$$\begin{aligned} \Vert A f\Vert _{{{\mathcal {X}}}}^2 \le C \Vert f\Vert _{{{\mathcal {X}}}} \big \Vert A^2 f\big \Vert _{{{\mathcal {X}}}}, \quad f \in {{\,\mathrm{dom}\,}}\big (A^2\big ), \end{aligned}$$

and recall that under exceedingly stronger hypotheses on the operator A and/or the Banach space \({{\mathcal {X}}}\), the optimal constant C in these inequalities diminishes from 4 (e.g., when A is the generator of a \(C_0\) contraction semigroup on a Banach space \({{\mathcal {X}}}\)) all the way down to 1 (e.g., when A is a symmetric operator on a Hilbert space \({{\mathcal {H}}}\)). We also survey some results in connection with an extension of the Hardy–Littlewood inequality involving quadratic forms as initiated by Everitt.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, 9th Printing. Dover, New York (1972)

    MATH  Google Scholar 

  2. Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Math, vol. 96. Birkhäuser, Basel (2001)

    Book  MATH  Google Scholar 

  3. Babenko, V.F.: Inequalities for norms of intermediate derivatives and some of their applications. In: Milovanovic, G.V. (ed.) Recent Progress in Inequalities, pp. 77–96. Springer, Kluwer, New York (1998)

    Chapter  Google Scholar 

  4. Behrndt, J., Hassi, S., De Snoo, H.: Boundary Value Problems, Weyl Functions, and Differential Operators, Monographs in Math, vol. 108. Springer, Birkhäuser, Cham (2020)

    Book  MATH  Google Scholar 

  5. Benammar, M., Bennewitz, C., Beynon, M.J., Brown, B.M., Dias, N.G.J., Evans, W.D., Everitt, W.N., Kirby, V.G., Littlejohn, L.L.: A catalogue of HELP and HELP-type integral and series inequalities. In: Milovanovic, G.V. (ed.) Recent Progress in Inequalities, pp. 127–160. Springer, Kluwer, New York (1998)

    Chapter  MATH  Google Scholar 

  6. Bennewitz, C.: A general version of the Hardy–Littlewood–Polya–Everitt (HELP) inequality. Proc. Roy. Soc. Edinb. 97A, 9–20 (1984)

    Article  MATH  Google Scholar 

  7. Berdyshev, V.I.: The best approximation in \(L_{[0,\infty )}\) of the operator of differentiation. Mat. Zametki 9, 477–481 (1971). (Russian.)

    MathSciNet  MATH  Google Scholar 

  8. Berens, H., Butzer, P.L., Westphal, U.: Representations of fractional powers of infinitesimal generators of semigroups. Bull. Am. Math. Soc. 74, 191–196 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bradley, J.S., Everitt, W.N.: On the inequality \(\Vert f^{\prime \prime }\Vert ^2 \le K \Vert f\Vert \big \Vert f^{(4)}\big \Vert \). Quart. J. Math. Oxf. 2(25), 241–252 (1974)

    Article  Google Scholar 

  10. Brodlie, K.W., Everitt, W.N.: On an inequality of Hardy and Littlewood. Proc. R. Soc. Edinb. 72A, 179–186 (1973/74)

  11. Brown, B.M., Evans, W.D., Everitt, W.N., Kirby, V.G.: Two integral inequalities. In: Agarwal, R.P. (ed.) Inequalities and Applications. World Scientific Series in Applicable Analysis, vol. 3, pp. 73–86. World Scientific, Singapore (1994)

    Chapter  MATH  Google Scholar 

  12. Brown, B.M., Kirby, V.G., Pryce, J.D.: Numerical determination of the Titchmarsh–Weyl \(m\)-coefficient and its applications to HELP inequalities. Proc. R. Soc. Lond. A 426, 167–188 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Certain, M.W., Kurtz, T.G.: Landau–Kolmogorov inequalities for semigroups and groups. Proc. Am. Math. Soc. 63, 226–230 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chernoff, P.R.: Optimal Landau–Kolmogorov inequalities for dissipative operators in Hilbert and Banach spaces. Adv. Math. 34, 137–144 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chisholm, R.S., Everitt, W.N.: On bounded integral operators in the space of integrable-square functions. Proc. R. Soc. Edinb. A 69, 199–204 (1970/71)

  16. Chisholm, R.S., Everitt, W.N., Littlejohn, L.L.: An integral operator inequality with applications. J. Inequalities Appl. 3, 245–266 (1999)

    MathSciNet  MATH  Google Scholar 

  17. Copson, E.T.: On two integral inequalities. Proc. R. Soc. Edinb. 77A, 325–328 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  18. Copson, E.T.: On two inequalities of Brodlie and Everitt. Proc. R. Soc. Edinb. 77A, 329–333 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  19. Davies, E.B.: One-Parameter Semigroups. Academic Press, London (1980)

    MATH  Google Scholar 

  20. Ditzian, Z.: Some remarks on inequalities of Landau and Kolmogorov. Aequationes Math. 12, 145–151 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  21. Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford Mathematical Monographs, 2nd edn. Oxford University Press, Oxford (2018)

    Book  MATH  Google Scholar 

  22. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Math, vol. 194. Springer, New York (2000)

    MATH  Google Scholar 

  23. Evans, W.D., Everitt, W.N.: A return to the Hardy–Littlewood integral inequality. Proc. R. Soc. Lond. A 380, 447–486 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  24. Evans, W.D., Everitt, W.N.: On an inequality of Hardy-Littlewood type: I, Proc. R. Soc. Edinb. 101A, 131–140 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  25. Evans, W.D., Everitt, W.N.: HELP inequalities for limit-circle and regular problems. Proc. R. Soc. Lond. A 432, 367–390 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  26. Evans, W.D., Everitt, W.N., Hayman, W.K., Jones, D.S.: Five integral inequalities; an inheritance from Hardy and Littlewood. J. Inequalities Appl. 2, 1–36 (1998)

    MathSciNet  MATH  Google Scholar 

  27. Evans, W.D., Everitt, W.N., Hayman, W.K., Ruscheweyh, S.: On a class of integral inequalities of Hardy–Littlewood type. J. Anal. Math. 46, 118–147 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  28. Evans, W.D., Zettl, A.: Norm inequalities involving derivatives. Proc. R. Soc. Edinb. 82A, 51–70 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  29. Everitt, W.N.: On an extension to an integro-differential inequality of Hardy, Littlewood and Polya. Proc. R. Soc. Edinb. 69A, 295–333 (1971)

    MathSciNet  MATH  Google Scholar 

  30. Everitt, W.N.: Integral inequalities and spectral theory. In: Everitt, W.N. (ed.) Spectral Theory and Differential Equations. Lecture Notes in Mathematics, vol. 448, pp. 148–166. Springer, Berlin (1975)

    Chapter  Google Scholar 

  31. Everitt, W.N.: A general integral inequality associated with certain ordinary differential operators. Quaest. Math. 2, 479–494 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  32. Everitt, W.N.: Analytic and numerical aspects of the HELP integral inequality. In: Sleeman, B.D., Jarvis, R.J. (eds.) Ordinary and Partial Differential Equations, vol. IV, pp. 19–48. Longman, Harlow (1993)

    Google Scholar 

  33. Everitt, W.N., Jones, D.S.: On an integral inequality. Proc. R. Soc. Lond. A 357, 271–288 (1977)

    Article  MATH  Google Scholar 

  34. Everitt, W.N., Zettl, A.: On a class of integral inequalities. J. Lond. Math. Soc. 2(17), 291–303 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  35. Fattorini, H.O.: Second Order Linear Differential Equations in Banach Spaces. North-Holland Mathematics Studies, vol. 108. North-Holland, Amsterdam (1985)

    MATH  Google Scholar 

  36. Franco, Z.M., Kaper, H.G., Kwong, M.K., Zettl, A.: Bounds for the best constants in Landau’s inequality on the line. Proc. R. Soc. Edinb. 95A, 257–262 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  37. Franco, Z.M., Kaper, H.G., Kwong, M.K., Zettl, A.: Best constants in norm inequalities for derivatives on a half-line. Proc. R. Soc. Edinb. 100A, 67–84 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  38. Gesztesy, F., Littlejohn, L.L., Michael, I., Wellman, R.: On Birman’s sequence of Hardy–Rellich–type inequalities. J. Differ. Eq. 264, 2761–2801 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gesztesy, F., Littlejohn, L.L., Michael, I., Pang, M.M.H.: A sequence of weighted Birman–Hardy–Rellich inequalities with logarithmic refinements, arxiv:2003.12894

  40. Gesztesy, F., Littlejohn, L.L., Nichols, R.: On self-adjoint boundary conditions for singular Sturm–Liouville operators bounded from below. J. Differ. Eq. 269, 6448–6491 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  41. Gindler, H.A., Goldstein, J.A.: Dissipative operator versions of some classical inequalities. J. Anal. Math. 28, 213–238 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  42. Gindler, H.A., Goldstein, J.A.: Dissipative operators and series inequalities. Bull. Aust. Math. Soc. 23, 429–442 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  43. Goldstein, J.A.: Semigroups of Linear Operators and Applications, 2nd edn. Oxford University Press, New York 1985. Dover Publications, New York (2017)

  44. Goldstein, J.A., Kwong, M.K., Zettl, A.: Weighted Landau inequalities. J. Math. Anal. Appl. 95, 20–28 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  45. Goldstein, J.A., Räbiger, F.: On Hardy–Landau–Littlewood inequalities. In: Semesterbericht Funktionalanalysis. Workshop on Operator Semigroups and Evolution Equations, Blaubeuren, Oct. 30–Nov. 3, 1989, Mathematisches Institut, Eberhard-Karls-Universität Tübingen, 1990, pp. 1–15; reprinted on p. 237 in the 2nd ed. of ref. [43]

  46. Hadamard, J.: Sur le module maximum d’une fonction et de ses dérivées. C.R. Acad. Sci. Paris Ser. A 42, 68–72 (1914)

    Google Scholar 

  47. Hardy, G.H., Landau, E., Littlewood, J.E.: Some inequalities satisfied by the integrals or derivatives of real or analytic functions. Math. Z. 39, 677–695 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hardy, G.H., Littlewood, J.E.: Some integral inequalities connected with the calculus of variations. Quart. J. Math. Oxford Ser. 2(3), 241–252 (1932)

    Article  MATH  Google Scholar 

  49. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952), reprinted (1988)

  50. Hayashi, M., Ozawa, T.: On Landau–Kolmogorov inequalities for dissipative operators. Proc. Am. Math. Soc. 145, 847–852 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  51. Hinton, D.: Limit point-limit circle criteria for \((p y^{\prime })^{\prime } + q y = \lambda k y\). In: Sleeman, B.D., Michael, I.M. (eds.) Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, vol. 415, pp. 173–183. Springer, Berlin (1974)

    Chapter  Google Scholar 

  52. Holbrook, J.A.R.: A Kallman–Rota inequality for nearly Euclidean spaces. Adv. Math. 14, 335–345 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  53. Kalf, H.: A characterization of the Friedrichs extension of Sturm–Liouville operators. J. Lond. Math. Soc. 2(17), 511–521 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  54. Kalf, H., Walter, J.: Strongly singular potentials and essential self-adjointness of singular elliptic operators in \(C_0^{\infty }({\mathbb{R}}^n \backslash \{0\})\). J. Funct. Anal. 10, 114–130 (1972)

    Article  MATH  Google Scholar 

  55. Kallman, R.R., Rota, G.-C.: On the inequality \(\Vert f^{\prime }\Vert ^2\le 4\Vert f\Vert / \Vert f^{\prime }\Vert \). In: Shisha, O. (ed.) Inequalities-II, pp. 187–192. Academic Press, New York (1970)

    Google Scholar 

  56. Kamke, E.: Differentialgleichungen. Lösungsmethoden und Lösungen. Gewöhnliche Differentialgleichungen, 7th edn. Akademische Verlagsgesellschaft, Leipzig (1961)

  57. Kato, T.: On an inequality of Hardy, Littlewood, and Pólya. Adv. Math 7, 217–218 (1971)

    Article  MATH  Google Scholar 

  58. Kato, T.: Perturbation Theory for Linear Operators, corrected printing of the 2nd edn. Springer, Berlin (1980)

  59. Kato, T., Satake, I.: An algebraic theory of Landau–Kolmogorov inequalities. Tôhoku Math. J. 33, 421–428 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  60. Kolmogorov, A.N.: On inequalities between upper bounds of the consecutive derivatives of an arbitrary function defined on an infinite interval, Uchenye Zapiski Moskov. Gos. Univ. Matematika 30, 3–16 (1939) (Russian). Am. Math. Soc. Trans. (1) 2, 233–243 (1962)

  61. Krasnosel’skii, M.A., Pustylnik, E.I., Sobolevskii, P.E., Zabreiko, P.P.: Integral Operators in Spaces of Summable Functions. Noordhoff International Publishing, Leyden (1976)

    Book  MATH  Google Scholar 

  62. Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy Inequality: About its History and Some Related Results. Vydavatelský Servis, Pilsen (2007)

    MATH  Google Scholar 

  63. Kupcov, N.P.: Kolmogorov estimates for derivatives in \(L_2[0,\infty )\). Proc. Steklov Inst. Math. 138, 101–125 (1975)

    MATH  Google Scholar 

  64. Kwong, M.K., Zettl, A.: Remarks on best constants for norm inequalities among powers of an operator. J. Approx. Theory 26, 249–258 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  65. Kwong, M.K., Zettl, A.: An extension of the Hardy–Littlewood inequality. Proc. Am. Math. Soc. 77, 117–118 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  66. Kwong, M.K., Zettl, A.: Norm inequalities for dissipative operators on inner product spaces. Houst. J. Math. 5, 543–557 (1979)

    MathSciNet  MATH  Google Scholar 

  67. Kwong, M.K., Zettl, A.: Ramifications of Landau’s inequality. Proc. R. Soc. Edinb. 86A, 175–212 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  68. Kwong, M.K., Zettl, A.: Norm inequalities for derivatives. In: Everitt, W.N., Sleeman, B.D. (eds.) Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, vol. 846, pp. 227–243. Springer, Berlin (1981)

    Chapter  Google Scholar 

  69. Kwong, M.K., Zettl, A.: Norm inequalities of product forms in weighted \(L^p\) spaces. Proc. R. Soc. Edinb. 89A, 293–307 (1981)

    Article  MATH  Google Scholar 

  70. Kwong, M.K., Zettl, A.: Landau’s inequality. Proc. R. Soc. Edinb. 97A, 161–163 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  71. Kwong, M.K., Zettl, A.: Norm inequalities for derivatives and differences. In: Everitt, W.N. (ed.) Inequalities. Fifty Years on from Hardy, Littlewood and Pólya, pp. 91–121. M. Dekker, New York (1991)

    Google Scholar 

  72. Kwong, M.K., Zettl, A.: Norm Inequalities for Derivatives and Differences. Lecture Notes in Mathematics, vol. 1536. Springer, New York (1992)

    MATH  Google Scholar 

  73. Kwong, M.K., Zettl, A.: An alternate proof of Kato’s inequality. In: Goldstein, G.R., Nagel, R., Romanelli, S. (eds.) Evolution Equations. Proceedings in Honor of J. A. Goldstein’s 60th Birthday, pp. 275–279. M. Dekker, New York (2003)

    Google Scholar 

  74. Landau, E.: Einige Ungleichungen für zweimal differenzierbare Funktionen. Proc. Lond. Math. Soc 13, 43–49 (1913)

    MATH  Google Scholar 

  75. Lizama, C., Miana, P.J.: A Landau–Kolmogorov inequality for generators of families of bounded operators. J. Math. Anal. Appl. 371, 614–623 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  76. Ljubič, Yu.I.: On inequalities between the powers of a linear operator. Izv. Akad. Nauk SSSR Ser. Mat. 24, 825–864 (1960) (Russian). Am. Math. Soc. Trans. (1) 40, 39–84 (1964)

  77. Martinez Carracedo, C., Sanz Alix, M.: The Theory of Fractional Powers of Operators. North-Holland Mathematics Studies, vol. 187. Elsevier, Amsterdam (2001)

  78. Muckenhoupt, B.: Hardy’s inequality with weights. Stud. Math. 44, 31–38 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  79. Naimark, M.A.: On the square of a closed symmetric operator. Dokl. Akad. Nauk SSSR 26, 863–867 (1940); 28, 207–208 (1940) (Russian)

  80. Neta, B.: On determination of best-possible constants in integral inequalities involving derivatives. Math. Comput. 35, 1191–1193 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  81. Niessen, H.-D., Zettl, A.: Singular Sturm–Liouville problems: the Friedrichs extension and comparison of eigenvalues. Proc. Lond. Math. Soc. 3(64), 545–578 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  82. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  MATH  Google Scholar 

  83. Phóng, V.Q.: On inequalities for powers of linear operators and for quadratic forms. Proc. R. Soc. Edinb. 89A, 25–50 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  84. Protter, M.H.: Inequalities for powers of unbounded operators. Manuscripta Math. 28, 71–79 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  85. Rassias, J.M.: Landau’s type inequalities. J. Math. Anal. Appl. 202, 280–301 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  86. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Fourier Analysis, Self-Adjointness, II. Academic Press, New York (1975)

    MATH  Google Scholar 

  87. Rosenberger, R.: A new characterization of the Friedrichs extension of semibounded Sturm–Liouville operators. J. Lond. Math. Soc. 2(31), 501–510 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  88. Schmüdgen, K.: On domains of powers of closed symmetric operators. J. Oper. Theory 9, 53–75 (1983)

    MathSciNet  Google Scholar 

  89. Siddiqi, J.A., Elkoutri, A.: Norm inequalities for generators of analytic semigroups and cosine operator functions. Canad. Math. Bull. 32, 47–53 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  90. Stečhkin, S.B.: Inequalities between norms of derivatives of arbitrary functions. Acta Sci. Math. (Szeged) 26, 225–230 (1965). (Russian)

    MathSciNet  Google Scholar 

  91. Stein, E.M.: Functions of exponential type. Ann. Math. 65, 582–592 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  92. Talenti, G.: Osservazioni sopra una classe di disuguaglianze. Rend. Sem. Mat. Fis. Milano 39, 171–185 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  93. Tomaselli, G.: A class of inequalities. Boll. Un. Mat. Ital. 4(2), 622–631 (1969)

    MathSciNet  MATH  Google Scholar 

  94. Trebels, W., Westphal, U.: A note on the Landau-Kallman-Rota-Hille inequality. In: Butzer, P.L., Kahane, J.-P., Szőkefalvi-Nagy, B. (eds.) Linear Operators and Approximation, vol. 20, pp. 115–119. Birkhäuser, Basel (1972)

    Chapter  Google Scholar 

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Acknowledgements

We are indebted to Man Kam Kwong and Lance Littlejohn for helpful discussions.

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  1. Department of Mathematics, Baylor University, Sid Richardson Bldg., 1410 S. 4th Street, Waco, TX, 76706, USA

    Fritz Gesztesy & Jonathan Stanfill

  2. Department of Mathematics (Dept. 6956), The University of Tennessee at Chattanooga, 615 McCallie Ave, Chattanooga, TN, 37403, USA

    Roger Nichols

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  1. Fritz Gesztesy
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Correspondence to Fritz Gesztesy.

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Communicated by Seppo Hassi.

Dedicated with great pleasure to Henk de Snoo on the happy occasion of his 75th birthday.

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This article is part of the topical collection “Recent Developments in Operator Theory - Contributions in Honor of H.S.V. de Snoo” edited by Jussi Behrndt and Seppo Hassi.

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Gesztesy, F., Nichols, R. & Stanfill, J. A Survey of Some Norm Inequalities. Complex Anal. Oper. Theory 15, 23 (2021). https://doi.org/10.1007/s11785-020-01060-9

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