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Criteria on boundedness of matrix operators in weighted spaces of sequences and their applications. (English) Zbl 1225.26056

Summary: In this paper we prove a new discrete Hardy-type inequality involving a kernel which has a more general form than those known in the literature.

We obtain necessary and sufficient conditions for the boundedness of a matrix operator from the weighted ${l}_{p,v}$ space into the weighted ${l}_{q,u}$ space defined by

${\left(Af\right)}_{j}:=\sum _{i=1}^{\infty }{a}_{i,j}{f}_{i},$

for all $f={\left\{{f}_{i}\right\}}_{i=1}^{\infty }\in {L}_{p,v}$ in case $1 and ${a}_{i,j}\ge 0$. Then we deduce a corresponding dual statement.

##### MSC:
 26D15 Inequalities for sums, series and integrals of real functions 47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)