On the Marchaud-type inequality

Marchaud’s inequality, which is valid in many function spaces, was strengthened by M. F. Timan [ 3 ] for L p , 1 > p > ∞ {L_p},1 > p > ∞ , 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 p and that translation is an isometry.