Weight characterizations for the discrete Hardy inequality with kernel

A discrete Hardy-type inequality is considered for a positive "kernel" , , and . For kernels of product type some scales of weight characterizations of the inequality are proved with the corresponding estimates of the best constant . A sufficient condition for the inequality to hold in the general case is proved and this condition is necessary in special cases. Moreover, some corresponding results for the case when are replaced by the nonincreasing sequences are proved and discussed in the light of some other recent results of this type.