On the Marchaud-type inequality

Ditzian, Z.

doi:10.1090/S0002-9939-1988-0938668-8

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

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.

References

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