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On an inequality of Kolmogorov and Stein

  • Ha Huy Bang (a1) and Hoang Mai Le (a2)
Abstract

A.N. Kolmogorov showed that, if f, f′, …, f (n) are bounded continuous functions on ℝ, then when 0 < k < n. This result was extended by E.M. Stein to Lebesgue Lp-spaces and by H.H. Bang to Orlicz spaces. In this paper, the inequality is extended to more general function spaces.

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References
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[1]Bang H.H., ‘A remark on the Kolomogorov–Stein inequality’, J. Math. Anal. Appl. 203 (1996), 861867.
[2]Bertrandias J.P. and Dupusi C., ‘Transformation de Fourier sur les espaces lp (Lp′)’, Ann. Inst. Fourier Grenoble 29 (1979), 189206.
[3]Bloom W.R., ‘Estimates for the Fourier transform’, Math. Scientist 10 (1985), 6581.
[4]Certain M.W. and Kurtz T.G., ‘Landau–Kolmogorov inequalities for semigroups and groups’, Proc. Amer. Math. Soc. 63 (1977), 226230.
[5]Fournier J.J.F., ‘On the Hausdorff–Young theorem for amalgams’, Monatsh. Math. 95 (1983), 117135.
[6]Holland F., ‘Harmonic analysis on amalgams of Lp and lq’, J. London Math. Soc. 10 (1975), 295305.
[7]Holland F., ‘On the representation of functions as Fourier transforms of unbounded measures’, Proc. London Math. Soc. 30 (1975), 347365.
[8]Kolmogorov A.N., ‘On inequalities between upper bounds of the successive derivatives of an arbitrary function on an infinite interval’, Uchen. Zap. Moskov. Gos. Uni. 30 (1939), 316.
[9]Kwong M.K. and Zettl A., Norm inequalities for derivatives and differences, Lecture Notes in Math. 1536 (Springer-Verlag, Berlin, Heidelberg, New York, 1992).
[10]Stein E.M., ‘Functions of exponential type’, Ann. Math. 65 (1957), 582592.
[11]Tikhomirov V.M. and Magaril-II'jaev G.G., ‘Inequalities for derivatives’, in Kolmogorov, A.N., Selected Papers (Nauka, Moscow, 1985).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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