Let $1\leq p <\infty$ 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)ds \bigg)^q u(t) dt \bigg)^{ \frac{ r}{ q}} w(x) dx \bigg)^{ \frac{ 1}{ r}} \leq C \bigg(\int_a^b f^p(x) v(x) dx \bigg)^{ \frac{ 1}{ p}} \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.
Categories: math.FA