Four questions related to Hardy's inequality (Q2712746)

The author studies four questions concerning various modifications of the Hardy integral inequality. First, he proves a lemma which enables one to reduce the question of the Hardy inequality on a hyperplane, namely NEWLINE\[NEWLINE \left(\int_{0}^{1}\left|\int_{0}^{x}f(t) dt\right|^qu(x) dx\right)^{1/q} \leq C \left(\int_{0}^{1}\left|f(t)\right|^pv(t) dt\right)^{1/p} NEWLINE\]NEWLINE for all \(f\) in a weighted Lebesgue space \(L^p (v)\) such that \(\int_{0}^{1}fm=0\) where \(m\) is a fixed function, to the usual weighted Hardy inequality. Next, he shows that the inequality NEWLINE\[NEWLINE \int_{0}^{1}\left|\int_{0}^{x}f(t) dt\right|^{q(x)}u(x) dx \leq C \int_{0}^{1}\left|f(t)\right|^{p(t)}v(t) dt NEWLINE\]NEWLINE can never hold unless the functions \(p(x)\) and \(q(x)\) are equal to the same constant. Third, the operator \(\int_{a(x)}^{b(x)}f(t) dt\) is studied. Weighted inequalities for this operator have been characterized by \textit{H. P. Heinig} and \textit{G. Sinnamon} [Stud. Math. 129, No. 2, 157-177 (1998; Zbl 0910.26008)] for \(a\), \(b\) non-decreasing and such that \(a(x)\leq b(x)\), and also by Gogatishvili and Lang (to appear) under the Berzhnoi's \(\ell\)-condition. A complete characterization is given when \(a\equiv 0\); in such case, the monotonicity restriction on \(b\) is removed. Finally, the inequality NEWLINE\[NEWLINE \left(\int_{0}^{1}\left(\int_{0}^{x}(x-t)^kf(t) dt\right)^q d\mu(x)\right)^{1/q} \leq C \left(\int_{0}^{1}f(t)^pv(t) dt\right)^{1/p} NEWLINE\]NEWLINE is considered, where \(v\) is a fixed weight. The author characterizes the class of measures \(\mu\) for which the above inequality holds.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00028].