The modular inequalities for a class of convolution operators on monotone functions

This paper is devoted to the study of modular inequality

where $\Phi _{1} \ll \Phi _{2}$ and $K$ is a class of Volterra convolution operators restricted to the monotone functions. When $\Phi _{1}(x) = x^{p}/p, \Phi _{2}(x) = x^{q}/q$ with $1 < p \le q < +\infty$ and the kernel $k(x) \equiv 1$, our results will extend those for the Hardy operator on monotone functions on Lebesgue spaces.