A new development of Nikol'skii - Lizorkin and Hardy type inequalities with applications

Abstract: This thesis consists of an introduction and five papers, which all deal with a new Sobolev type function space called the space with multiweighted derivatives. As basis for this space serves some differential operators containing weight functions. In the introduction we present the reasons why this operator appears naturally and also point out some possible application areas. In the first and the second papers we present and investigate a different way to characterize the behavior of a function from this space at the singular point zero. The main goal of these papers is to find suitable conditions for the validity of a Nikol'skii - Lizorkin type inequality for functions in this space. This inequality, in turn, is a generalization of the Poincare and Friedrichs inequalities, and it can be applied to the solution of elliptic boundary value problems because it involves the estimation of a function via its higher order derivatives and non-homogenous boundary values. In the first and the second papers we consider different classes of boundary values. The third and the fourth papers are devoted to a special generalization of the higher order Hardy inequality. The generalization consists of considering our special differential operator instead of a higher order derivative. Moreover, in the fourth paper the proofs of the main results face with the problem to characterize a new Hardy type inequality (for a Volterra type operator), which is of independent interest. In the fifth paper we study spectral properties of some differential operators by using a special technique based on Hardy type inequalities. Here, in particular, we use in a crucial way the results concerning the new Hardy type inequalities we proved in the third and the fourth papers.