Some new boundedness and compactness results for discrete Hardy type operators with kernels

University dissertation from Luleå : Luleå tekniska universitet

Abstract: This thesis consists of an introduction and three papers, which deal with some new discrete Hardy type inequalities. In the introduction we give an overview of the area that serves as a frame for the rest of the thesis. In particular, a short description of the development of Hardy type inequalities is given.In Paper 1 we prove a new discrete Hardy-type inequality $$ \|Af\|_{q,u}\leq C\|f\|_{p,v},~~~~1$$where the matrix operator $A$ is defined by $\left(Af\right)_i:=\sum\limits_{j=1}^ia_{i,j}f_j,$ ~$a_{i, j}\geq 0$, where the entries $a_{i, j}$ satisfies less restrictive additional conditions than studied before. Moreover, we study the problem of compactness for the operator $A$, and also the dual result is stated, proved and discussed.In Paper 2 we derive the necessary and sufficient conditions for inequality (1) to hold for the case $1 In Paper 3 we consider an operator of multiple summation with weights in weighted sequence spaces, which cover a wide class of matrix operators and we state, prove and discuss both boundedness and compactness forthis operator, for the case $1

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