We prove a new discrete Hardy-type inequality A f q,u a i, j f j , a i, j 0 . Moreover, we study the problem of compactness of the operator A , and the dual result is stated. (2000): 26D10, 26D15, 47B37. Mathematics subject classification

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Abstract. We study the problem of boundedness and compactness of operators of multiple summation with weights in weighted sequence spaces.Mathematics subject classification (2000): 39B82, 44B20, 46C05.

Abstract. We derive necessary and sufficient conditions (of Muckenhoupt-Bradley type) for the validity of q -analogs of (r, p) -weighted Hardy-type inequalities for all possible positive values of the parameters r and p . We also point out some possibilities to further develop the theory of Hardy-type inequalities in this new direction.Mathematics subject classification (2010): 26D10, 26D15, 33D05, 39A13.