Overdetermined weighted Hardy inequalities on semiaxis. (English) Zbl 1006.26014

Nasyrova, Maria

Edmunds, David E. (ed.) et al., Function spaces and applications. Mainly the proceedings of the international conference on function spaces and applications to partial differential equations, New Delhi, India, December 15-19, 1997. New Delhi: Narosa Publishing House. 201-231 (2000).

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 [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 M. Nasyrova and 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.
For the entire collection see [Zbl 0958.00028].

Reviewer: Luboš Pick (Praha)

MSC:

26D10	Inequalities involving derivatives and differential and integral operators
46N20	Applications of functional analysis to differential and integral equations
47G10	Integral operators

Keywords:

overdetermined weighted Hardy inequality; zeros at both ends

Citations:

Zbl 0837.26012; Zbl 0894.26007