THE MODULAR INEQUALITIES FOR A CLASS OF CONVOLUTION-OPERATORS ON MONOTONE-FUNCTIONS

This paper is devoted to the study of modular inequality Phi(2)(-1) (i ntegral(0)(+infinity) Phi(2)(a(x)Kf(x))w(x)dx) less than or equal to P hi(1)(-1) (integral(0)(+infinity) Phi(1)(Cf(x))upsilon(x)dx) where Phi (1) much less than Phi(2) and K is a class of Volterra convolution ope rators restricted to the monotone functions. When Phi(1)(x) = x(p)/p, Phi(2)(x) = x(q)/q with 1 < p less than or equal to q < +infinity and the kernel k(x) = 1, our results will extend those for the Hardy opera tor on monotone functions on Lebesgue spaces.