Lower bounds of Copson type for Hausdorff matrices II

Abstract

Let A = ( a n, k ) n, k⩾0 be a non-negative matrix. Denote by L p, q ( A) the supremum of those L satisfying the following inequality: ∑ n = 0 ∞ ∑ k = 0 ∞ a n , k x k q 1 / q ⩾ L ∑ k = 0 ∞ x k p 1 / p ( X ∈ ℓ p , X ⩾ 0 ) . The purpose of this paper is to establish a Hardy-type formula for L p, q ( H μ ), where H μ is a Hausdorff matrix and 0 < q ⩽ p ⩽ 1. A similar result is also established for L p , q ( H μ t ) with −∞ < q ⩽ p < 0. As a consequence, we apply them to Cesàro matrices, Hölder matrices, Gamma matrices, generalized Euler matrices, and Hausdorff matrices with monotone rows. Our results fill up the gap which the work of Bennett has not dealt with.