In this paper, we prove some reverse discrete inequalities with weights of Muckenhoupt and Gehring types and use them to prove some higher summability theorems on a higher weighted space $l_{w}^{p}({\open N})$ form summability on the weighted space $l_{w}^{q}({\open N})$ when p>q. The proofs are obtained by employing new discrete weighted Hardy's type inequalities and their converses for non-increasing sequences, which, for completeness, we prove in our special setting. To the best of the authors' knowledge, these higher summability results have not been considered before. Some numerical results will be given for illustration.

中文翻译：

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更高的可和性和离散加权 Muckenhoupt 和 Gehring 类型不等式

在本文中，我们证明了一些具有 Muckenhoupt 和 Gehring 类型权重的反向离散不等式，并用它们证明了更高权重空间上的一些更高可和性定理$l_{w}^{p}({\open N})$在加权空间上形成可和性$l_{w}^{q}({\open N})$什么时候p>q. 证明是通过采用新的离散加权哈代型不等式及其对非增序列的逆来获得的，为了完整起见，我们在特殊设置中证明了这一点。据作者所知，这些更高的可总结性结果以前没有被考虑过。为了说明，将给出一些数值结果。