Lower bounds for generalized Hausdorff matrices and lower triangular matrices on the block weighted sequence space ?_p(w, F)

摘要

Let 0 < p < 1 and H = (h_(n,k))_(n,k≥1) be a non-negative matrix. Denote by L_(w,p,q,F) (H), the supremum of those L, satisfying the following inequality where F = (F_n) is a partition of positive integers, I_n = {n}, x is a non-negative sequence in ?_p(w, I) and (w_n) is a monotone and non-negative sequence of real number. In this paper a Hardy-type formula is obtained for L_(w,p,q,F) (H_μ~α), where H_μ~α is the generalized Hausdorff matrix, 0 < q ≤ p < 1 and α > 0. Another purpose of this paper is to establish a general upper estimate for the exact value of L_(w,p,I) (H~t), for which recently a lower estimate was established in Lashkaripour and Talebi [Lashkaripour R, Talebi G. Bull. Iran. Math. Soc. 2011;37:115-126], where H is a non-negative lower triangular matrix and 0 < p < 1. We also derive the corresponding result for L_(w,p,I) (H), with ?∞ < p < 0. In particular, we apply our results to summability matrices, weighted mean matrices, N?rlund matrices. Our results also generalize some results in Chen and Wang [Chen C-P, Wang K-Z. Linear Multilinear Algebra, March 2011;59:321-337] and Lashkaripour and Talebi [Lashkaripour R, Talebi G. Czech. Math. J. 2012; 62: 293-04.].