Lower bound and upper bound of operators on block weighted sequence spaces

Abstract

Let $A = {({a_{n,k}})_{n,k \ge 1}}$ be a non-negative matrix. Denote by L _v,p,q,F (A) the supremum of those L that satisfy the inequality

$$\parallel Ax{\parallel _{v,q,F}} \ge L\parallel x{\parallel _{v,p,F}}$$

where x ⩾ 0 and x ε ℓ _p (v, F) and also v = (v _n ) ^∞ _n=1 is an increasing, non-negative sequence of real numbers. If p = q, we use L _v,p,F (A) instead of L _v,p,p,F (A). In this paper we obtain a Hardy type formula for L _v,p,q,F (${H_\mu }$), where ${H_\mu }$ is a Hausdorff matrix and 0 < q ⩽ p ⩽ 1. Another purpose of this paper is to establish a lower bound for ‖A ^NM _W ‖ _v,p,F , where A ^NM _W is the Nörlund matrix associated with the sequence W = {w _n } ^t8 _n=1 and 1 < p < ∞. Our results generalize some works of Bennett, Jameson and present authors.