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Let $T_{\gamma }f\left(x\right)={ʃ}_0^{x}k{(x,y)}^{\gamma }f\left(y\right)dy$, where k is a nonnegative kernel increasing in x, decreasing in y, and satisfying a triangle inequality. An nth-order Opial inequality has the form ${ʃ}_0^{\infty }({\prod }_{i=1}^n|T_{{\gamma }_i}{f\left(x\right)|}^{q_i}{\left|\right)\left|f\left(x\right)\right|}^{q_0}w\left(x\right)dx\le C({ʃ}_0^{\infty }{\left|f\left(x\right)\right|}^p{v\left(x\right)dx)}^{(q_0+\dots +q_n)/p}$. Such inequalities can always be simplified to nth-order reduced inequalities, where the exponent $q_0=0$. When n = 1, the reduced inequality is a standard weighted norm inequality, and characterizing the weights is easy. We also find necessary and sufficient conditions on the weights for second-order reduced Opial inequalities to hold.

Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for ${\Phi }_2^{-1}(ʃ{\Phi }_2\left(w\left(x\right)\right|Tf\left(x\right)\left|\right)t\left(x\right)dx)\le {\Phi }_1^{-1}(ʃ{\Phi }_1\left(Cu\left(x\right)\right|f\left(x\right)\left|\right)v\left(x\right)dx)$ to hold when ${\Phi }_1$ and ${\Phi }_2$ are N-functions with ${\Phi }_2\circ {\Phi }_1^{-1}$ 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.