First and second order Opial inequalities

Let ${\displaystyle T_{\gamma }f\left(x\right)={ʃ}_0^{x}k{(x,y)}^{\gamma }f\left(y\right)dy}$, 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 ${\displaystyle {ʃ}_0^{\infty }\left({\prod }_{i=1}^n{\left|T_{{\gamma }_i}f\left(x\right)\right|}^{q_i}\mid \right){\left|f\left(x\right)\right|}^{q_0}w\left(x\right)dx\le C{\left({ʃ}_0^{\infty }{\left|f\left(x\right)\right|}^{p}v\left(x\right)dx\right)}^{\frac{q_0+\dots +q_n}{p}}}$. Such inequalities can always be simplified to nth-order reduced inequalities, where the exponent ${\displaystyle q_0=0}$. 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.