Weighted inequalities involving iteration of two Hardy integral operators

Let $0 < p \leq 1$ and $0 < q,r < \infty$. We characterize validity of the inequality for the composition of the Hardy operator, \begin{equation*} \bigg(\int_a^b \bigg(\int_a^x \bigg(\int_a^t f(s)^p v(s) ds \bigg)^\frac{q}{p} u(t) dt \bigg)^{\frac{r}{q}} w(x) dx \bigg)^{\frac{1}{r}} \leq C \int_a^b f(x) dx, \end{equation*} for all non-negative measurable functions on $(a,b)$, $-\infty \leq a < b \leq \infty$. We construct a more straightforward discretization method than those previously presented in the literature, and we characterize this inequality in both discrete and continuous forms.