Lower bounds of Copson type for the transposes of lower triangular matrices

Let A = (an,k)n,k 0 be a non-negative matrix. Denote by Lp,q(A) the supremum of those L satisfying the following inequality: ∞ n=0 ∞ k=0 an,kxk q 1/q L ∞ k=0 xk p 1/p (X ∈ p, X 0). In this paper, we focus on the evaluation of Lp,p(At ) for a lower triangular matrix A, where 0 < p <1. A Borwein-type result is established. We also derive the corresponding result for the case Lp,p(A) with −∞ < p <0. In particular, we apply them to summability matrices, the weighted mean matrices, and Nörlund matrices. Our results not only generalize the work of Bennett, but also provide several analogues of those given in [Chang-Pao Chen, Dah-Chin Lour, Zong-Yin Ou, Extensions of Hardy inequality, J. Math. Anal. Appl. 273 (1) (2002) 160–171] and [P.D. Johnson Jr., R.N. Mohapatra, D. Ross, Bounds for the operator norms of some Nörlund matrices, Proc. Amer. Math. Soc. 124 (2) (1996), Corollary on p. 544]. Our results also improve Bennett’s results for some cases. © 2007 Elsevier Inc. All rights reserved