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On the Marchaud-type inequality

Author: Z. Ditzian
Journal: Proc. Amer. Math. Soc. 103 (1988), 198-202
MSC: Primary 26A15; Secondary 26D10, 41A25
MathSciNet review: 938668
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Abstract: 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.

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Keywords: Marchaud type inequality, moduli of smoothness, norm modulus of smoothness of power type $ p$
Article copyright: © Copyright 1988 American Mathematical Society

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