A generalization of Hardy’s inequality to infinite tensors

In this paper, we extend Hardy’s inequality to infinite tensors. To do so, we introduce Cesàro tensors ℭ {\mathfrak{C}} , and consider them as tensor maps from sequence spaces into tensor spaces. In fact, we prove inequalities of the form ∥ ℭ ⁢ x k ∥ t , 1 ≤ U ⁢ ∥ x ∥ l p k \|\mathfrak{C}x^{k}\|_{t,1}\leq U\|x\|_{l_{p}}^{k} ( k = 1 , 2 k=1,2 ), where x is a sequence, ℭ ⁢ x k {\mathfrak{C}x^{k}} is a tensor, and ∥ ⋅ ∥ t , 1 {\|\cdot\|_{t,1}} , ∥ ⋅ ∥ l p {\|\cdot\|_{l_{p}}} are the tensor and sequence norms, respectively. The constant U is independent of x, and we seek the smallest possible value of U.