### First and second order Opial inequalities

Let ${T}_{\gamma}f\left(x\right)={\u0283}_{0}^{x}k{(x,y)}^{\gamma}f\left(y\right)dy$, 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 ${\u0283}_{0}^{\infty}({\prod}_{i=1}^{n}|{T}_{{\gamma}_{i}}{f\left(x\right)|}^{{q}_{i}}{\left|\right)\left|f\left(x\right)\right|}^{{q}_{0}}w\left(x\right)dx\le C({\u0283}_{0}^{\infty}{\left|f\left(x\right)\right|}^{p}{v\left(x\right)dx)}^{({q}_{0}+\dots +{q}_{n})/p}$. 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.