Weighted ${\displaystyle L_{\Phi }}$ integral inequalities for operators of Hardy type

Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for ${\displaystyle {\Phi }_2^{-1}(ʃ{\Phi }_2\left(w\left(x\right)\left|Tf\left(x\right)\right|\right)t\left(x\right)dx)\le {\Phi }_1^{-1}(ʃ{\Phi }_1\left(Cu\left(x\right)\left|f\left(x\right)\right|\right)v\left(x\right)dx)}$ to hold when ${\displaystyle {\Phi }_1}$ and ${\displaystyle {\Phi }_2}$ are N-functions with ${\displaystyle {\Phi }_2\circ {\Phi }_1^{-1}}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.