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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the Marchaud-type inequality
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by Z. Ditzian PDF
Proc. Amer. Math. Soc. 103 (1988), 198-202 Request permission


Marchaud’s inequality, which is valid in many function spaces, was strengthened by M. F. Timan [3] for ${L_p},1 < p < \infty$, following a technique of A. Zygmund [5]. In both the above-mentioned articles the powerful Littlewood-Paley theorem is used. In the present paper a direct and, I believe much simpler, proof is given for that stronger Marchaud-type inquality. Moreover, the result will apply to a more general class of function spaces. It will be shown that it is sufficient that the "modulus of smoothness" of the norm is of "power-type" $p$ and that translation is an isometry.
  • Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
  • Robert C. Sharpley, Cone conditions and the modulus of continuity, Second Edmonton conference on approximation theory (Edmonton, Alta., 1982) CMS Conf. Proc., vol. 3, Amer. Math. Soc., Providence, RI, 1983, pp. 341–351. MR 729339
  • M. F. Timan, Inverse theorems of the constructive theory of functions in $L_{p}$ spaces $(1\leq p\leq \infty )$, Mat. Sb. N.S. 46(88) (1958), 125–132 (Russian). MR 0100198
  • A. F. Timan, Theory of approximation of functions of a real variable, A Pergamon Press Book, The Macmillan Company, New York, 1963. Translated from the Russian by J. Berry; English translation edited and editorial preface by J. Cossar. MR 0192238
  • A. Zygmund, A remark on the integral modulus of continuity, Univ. Nac. Tucumán. Revista A. 7 (1950), 259–269. MR 0042479
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 103 (1988), 198-202
  • MSC: Primary 26A15; Secondary 26D10, 41A25
  • DOI:
  • MathSciNet review: 938668