Characterization of Lvp[0, ∞) - L μq[O, ∞) boundedness of the general Hardy operator (Hsf)(x) =(∫[0,x] fsudλ) 1/s restricted to monotone functions f ≥ 0 for 0 < p.q.s < ∞ with positive Borel σ -finite measures λ, μ and v is obtained.

A new scale of characterizations for the weighted Hardy inequality with variable limits is proved for the case 1 < p ≤ q < ∞. A corresponding scale of characterizations for the (limit) weighted Pólya-Knopp inequality is also derived and discussed.

Some Hardy-type integral inequalities in general measure spaces, where the corresponding Hardy operator is replaced by a more general Volterra type integral operator with kernel k(x,y), are considered. The equivalence of such inequalities on the cones of non-negative respective non-increasing functions are established and applied.

The conditions are investigated, under which for all Lebesgue measurable functions f(x) greater than or equal 0 on a semi-axis R+:=(0, infinity ) with a constant C greater than or equal 0 independent of f, satisfied is inequality: {0 integral infinity [Kf(x)]qν(x)dx}1/q [less-than or equal to] C{0 integral infinity [f(x)]pu(x)dx}1/p (1) with measurable weighted functions u(x) greater than or equal 0 and ν(x) greater than or equal 0 and integral operator Kf(x):=0 integral infinity k(x,y)f(y)dy, where measurable in R+×R+ kernel k(x,y) greater than or equal 0 is monotone in one or two variables. Such operators can be exemplified with Laplace, Hilbert transforms etc. Further, the comparison theorems for (1)-type inequalities with the similar inequalities on a cone of non-growing functions for certain-type Volterra operators are proved.

This PhD thesis consists of an introduction and six papers. All these papers are devoted to Lebesgue norm inequalities with Hardy type integral operators. Three of these papers also deal with so-called Pólya-Knopp type inequalities with geometric mean operators instead of Hardy operators. In the introduction we shortly describe the development and current status of the theory of Hardy type inequalities and put the papers included in this PhD thesis into this frame. The papers are conditionally divided into three parts. The first part consists of three papers, which are devoted to weighted Lebesgue norm inequalities for the Hardy operator with both variable limits of integration. In the first of these papers we characterize this inequality on the cones of non-negative monotone functions with an additional third inner weight function in the definition of the operator. In the second and third papers we find new characterizations for the mentioned inequality and apply the results for characterizing the weighted Lebesgue norm inequality for the corresponding geometric mean operator with both variable limits of integration. The second part consists of two papers, which are connected to operators with monotone kernels. In the first of them we give criteria for boundedness in weighted Lebesgue spaces on the semi-axis of certain integral operators with monotone kernels. In the second one we consider Hardy type inequalities in Lebesgue spaces with general measures for Volterra type integral operators with kernels satisfying some conditions of monotonicity. We establish the equivalence of such inequalities on the cones of non-negative respective non-increasing functions and give some applications. The third part consists of one paper, which is devoted to multi-dimensional Hardy type inequalities. We characterize here some new Hardy type inequalities for operators with Oinarov type kernels and integration over spherical cones in n-dimensional vector space over R. We also obtain some new criteria for a weighted multi-dimensional Hardy inequality (of Sawyer type) to hold with one of two weight functions of product type and give as applications of such results new characterizations of some corresponding n-dimensional weighted Pólya-Knopp inequalities to hold.