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

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.

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 |