Weight characterizations of discrete Hardy and Carleman type inequalities

Abstract: This thesis deals with some generalizations of the discrete Hardy and Carleman type inequalities and the relations between them. In Chapter 1 we give an introduction and overview of the area that serves as a frame for the rest of the thesis. In particular, a fairly complete description of the development of discrete Hardy and Carleman type inequalities in one and more dimensions can be found in this chapter. In Chapter 2 we consider some scales of weight characterizations for the one-dimensional discrete Hardy inequality for the case 1In Chapter 3 we present and discuss a new scale of weight characterizations for a two-dimensional discrete Hardy type inequality and its limit two-dimensional Carleman type inequality. In Chapter 4 we generalize the work done in Chapters 2 and 3 and present, prove and discuss the corresponding general n-dimensional versions. In Chapter 5 we introduce the study of the general Hardy type inequality with kernels involved. For kernels of product type a weight characterization is given, thus generalizing a previous result of M. Goldman. A scale of sufficient conditions is proved for the general case. Finally, in the Appendix some steps in the historical development of the continuous Hardy inequality are briefly described.