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

Abstract:

This paper is devoted to the study of modular inequality \begin{equation*}\Phi _{2}^{-1} \left ( \int _{0}^{+\infty } \Phi _{2}( a(x)K f(x)) w(x) dx \right ) \le \Phi _{1}^{-1} \left ( \int _{0}^{+\infty } \Phi _{1}( C f(x) ) v(x) dx \right ) \end{equation*} 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.