Overdetermined weighted Hardy inequalities on semiaxis (Q2712743)

The author considers the weighted Hardy inequality \(\|Fu\|_L^q(0,\infty) \leq C\|F^{(k)}v\|_{L^p(0,\infty)}\) with \(1<p,q<\infty\) for functions which are sufficiently differentiable and vanishing at both the endpoints together with their derivatives up to the order \(k-1\). This problem has been studied since late 1980's, but mainly on a finite interval, or under some further restrictions on the weights involved. The author's approach consists of turning the differential inequality to an equivalent one involving the Riemann-Liouville integral operator for which boundedness criteria are known [\textit{V. D. Stepanov}, J. Lond. Math. Soc., II. Ser. 50, No. 1, 105-120 (1994; Zbl 0837.26012)]. The paper extends the previous work by \textit{M. Nasyrova} and \textit{V. Stepanov} [J. Inequal. Appl. 1, No. 3, 223-238 (1997; Zbl 0894.26007)], in particular employing the ``heuristic principle'' developed in the cited paper. The paper brings a full exposition of results in all possible situations in the case when \(k=2\) and also a number of results concerning higher order inequalities. The results, naturally, contain also the best constant relations.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00028].