Weighted inequalities of Hardy-type on amalgams. (English) Zbl 1189.26033

Jain, Pankaj; Kumar, Suket

Real Anal. Exch. 34(2008-2009), No. 2, 483-499 (2009).

The aim of the authors is to characterize weighted Hardy-type inequalities between suitable amalgams \(l^q(L^p,u)\) and \(l^{\bar{q}}(L^{\bar{q}},u)\).
First, they characterize the boundedness of the operator \(T = T_1 + T_2\), where \(T_1 \) and \(T_2\) are discrete Hardy operators, not necessarily non-negative, between suitable weighted sequence spaces.
We recall that for the continuous case, the operator \(T\) has been studied by P. A. Zharov [Proc. Steklov Inst. Math. 194, 101–114 (1993); translation from Tr. Mat. Steklova 194, 97–110 (1992; Zbl 0811.26007)], or by A. Kufner and L.-E. Persson [“Weighted inequalities of Hardy type”, Singapore: World Scientific (2003; Zbl 1065.26018)].
Next, the authors study the boundedness of operators of the kind \(S=S_1+S_2\) associated to general functions not necessarily non-negative.
The boundedness of the operators \(S_1\) and \(S_2\) in some special case has been studied by Carton-Lebrun, Heinig and Hofmann, while in a more general case but for non-negative related functions, the \(L^p\)-\(L^q\) boundedness of \(S_1\) and \(S_2\) has been obtained by Stephanov.

Reviewer: Maria Alessandra Ragusa (Catania)

MSC:

26D10	Inequalities involving derivatives and differential and integral operators
26D15	Inequalities for sums, series and integrals

Keywords:

Hardy inequality; discrete Hardy inequalities; amalgam; Hardy operators; involving general functions.

Citations:

Zbl 0811.26007; Zbl 1065.26018