Best Possible Results in a Class of Inequalities, II

We give a sufficient condition on a lower triangular infinite matrix A with nonnegative entries, and a positive sequence b = (bn), for an inequality of the form ||A(b|x|)||p ≤ K||x||p, x ∈ ℓp, to be best possible, in the sense that there is no positive sequence d = (dn) such that (dnb−1n) is a monotone unbounded sequence, and an inequality of the form above holds with b replaced by d. This condition permits easy proofs of "best possible" theorems that generalize a previous result concerning Hardy′s inequality.