Extensions of Hardy inequality

AbstractLet 1<p<∞ and A=(an,k)n,k⩾0. Denote by ‖A‖p,p the number whose p-power is the infimum of those U satisfying the following inequality: ∑n=0∞∑k=0∞an,kxkp⩽U∑k=0∞|xk|pX≡{xn}n=0∞∈ℓp. The purpose of this paper is to give an upper bound and a lower bound for ‖A‖p,p. Our results not only generalize results of Bennett, Borwein and Johnson et al., but also improve the ones of Bennett and Borwein and Cass. We also give a partial answer to Problem 7.23 in Quart. J. Math. Oxford (2) 49 (1998) 395–432