Lower bound of four-dimensional Hausdorff matrices

Abstract

Let mathsf{H}=(h_{nmjk}) be a non-negative four-dimensional matrix. Denote by L_{p}(mathsf{H}) the supremum of those ℓ satisfying the following inequality: (∑n=0∞∑m=0∞(∑j=0∞∑k=0∞hnmjkxj,k)p)1/p≥ℓ(∑j=0∞∑k=0∞xj,kp)1/p,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ { \\Biggl( {\\sum_{n = 0}^{\\infty }{\\sum _{m = 0}^{\\infty } {{{ \\Biggl( {\\sum _{j = 0}^{\\infty }{\\sum_{k = 0}^{\\infty }{{h_{nmjk}} {x_{j,k}}} } } \\Biggr)}^{p}}} } } \\Biggr)^{1/p}} \\ge \\ell { \\Biggl( {\\sum_{j = 0}^{\\infty }{\\sum _{k = 0}^{ \\infty }{x_{j,k}^{p}} } } \\Biggr)^{1/p}}, $$\\end{document} where x=(x_{j,k}) in mathcal{L}_{p} with x_{j,k}ge 0. In this paper a Hardy type formula is established for L_{p}(mathsf{H}_{ mu times lambda }^{t}), where 0< ple 1 and mathsf{H}_{mu times lambda } is a four-dimensional Hausdorff matrix. A similar result is also obtained for the case in which mathsf{H}_{mu times lambda } is replaced by mathsf{H}_{mu times lambda }^{t}. As a consequence, we apply the results to some special four-dimensional Hausdorff matrices such as Cesàro, Euler, Hölder and Gamma matrices. Our results contain some generalizations of Copson’s discrete inequality.