Estimates for Discrete Hardy-type Operators in Weighted Sequence Spaces

Abstract: This PhD thesis consists of an introduction and eight papers, which deal with questions of the validity of some new discrete Hardy type inequalities in weighted spaces of sequences and on the cone of non-negative monotone sequences, and their applications. 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 find necessary and sufficient conditions on weighted sequences $\omega_i$, $i=1, 2,...,n-1$, $u$ and $v$, for which the operator $$ (S_{n}f)_i=\sum\limits_{k_1=1}^i\omega_{1,k_1}\cdots\sum\limits_{k_{n-1}=1}^{k_{n-2}} \omega_{n-1,k_{n-1}}\sum\limits_{j=1}^{k_{n-1}}f_j,~~ i\geq 1,~~~~~(1) $$ is bounded from $l_{p,v}$ to $l_{q,u}$ for $1<p\leq q<\infty$. In Paper 2 we prove a new discrete Hardy-type inequality $$ \|Af\|_{q,u}\leq C\|f\|_{p,v},~~~~1<p\leq q<\infty,~~~~~~~~~~~(2) $$ 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 $a_{i, j}$ satisfy 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 3 we derive the necessary and sufficient conditions for inequality (2) to hold for the case $1<q<p<\infty$. In Papers 4 and 5 we obtain criteria for the validity of the inequality (2) for slightly more general classes of matrix operators $A$ defined by $\left(Af\right)_j:=\sum\limits_{i=j}^\infty a_{i,j}f_i,$ ~$a_{i, j}\geq 0$, when $1<p, q<\infty$. Moreover, we study the problem of compactness for the operator $A$ for the case $1<p\leq q<\infty$, also the dual result is established and here we also give some applications of the main results. In Paper 6 we state boundedness for the operator of multiple summation with weights (1) in weighted sequence spaces for the case $1<q<p<\infty$. Paper 7 deals with new Hardy-type inequalities restricted to the cones of monotone sequences. The case $1<q<p<\infty$ is considered. Also some applications related to H\"{o}lder's summation method are pointed out. In Paper 8 we obtain necessary and sufficient conditions for which three-weighted Hardy type inequalities hold.