Criteria on boundedness of matrix operators in weighted spaces of sequences and their applications.

*(English)*Zbl 1225.26056Summary: 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<q<p<\infty $ and ${a}_{i,j}\ge 0$. Then we deduce a corresponding dual statement.