Inequalities involving dual of some averaging operators (Q2754358)

This paper deals with the weighted Hardy-type inequalities involving the dual operators NEWLINE\[NEWLINE\Big(H_{\alpha}f\Big)(x)= \begin{cases} \left(\frac 1{x}\int^{\infty}_x f^{\alpha}(t) dt\right)^{\frac 1{\alpha}}, &\text{if }\alpha \neq 0\\ \exp\left(\frac 1{x}\int^{\infty}_x \ln f(t) dt\right), &\text{if }\alpha = 0. \end{cases}NEWLINE\]NEWLINE In it, the authors give the characterization of the weight functions \(w\), \(v\) in the index range \(0<\alpha \leq p \leq q \leq \infty\) for which the inequality NEWLINE\[NEWLINE\|H_{\alpha}f\cdot w^{\frac 1{q}}\|_{q,(0,\infty)}\leq C\|f\cdot v^{\frac 1{p}}\|_{p,(0,\infty)}NEWLINE\]NEWLINE holds for all positive functions \(f\). Indeed, their results extend many known generalizations of Hardy-type inequalities given in [\textit{B. Opic} and \textit{A. Kufner}: ``Hardy-type inequalities'' (1990; Zbl 0698.26007)].NEWLINENEWLINEFor the entire collection see [Zbl 0971.00008].