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

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 ? _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 _W ^NM ‖ _v,p,F , where A _W ^NM is the N?rlund matrix associated with the sequence W = {w _n } _n=1 ^t8 and 1 < p < ∞. Our results generalize some works of Bennett, Jameson and present authors.