The complete characterization of the Hardy-type Lp-Lq inequalities on the weighted cones of quasi-concave functions for all 0 < p, q < ∞ is given.

The complete characterization of the weighted L^p− L^r inequalities of supremum operators on the cones of monotone functions for all 0 < p, r≤ ∞ is given.

This PhD thesis is devoted to the study weighted Hardy-type inequalitieswith quasilinear integral operators on the cone of monotone functions. Thethesis consists of six papers (papers A - F) and an introduction, which givesa brief review of the theory of Hardy-type inequalities and also serves to putthese papers into a more general frame.

In papers A, D and E we characterize some weighted Hardy-type inequal-ities on the cone of non-increasing functions. This problem is related to theboundedness of the Hardy-Littlewood maximal operator in weighted LorentzΓ - spaces. In papers D and E the case with integral operators defined byso called Oinarov’s kernels are treated. In all cases necessary and sufficientconditions are derived.

In paper B we solve the similar problem for the cone of quasi-concavefunctions (i.e. when the function f satisfy two monotonicity conditions,namely that f (t) is non-decreasing and f(t)t is non-increasing). Such functions are of great importance for interpolation theory, approximation theory and related areas in functional analysis. Also here complete characterizations are given in all cases.

Paper C is devoted to characterizing weighted Hardy-type inequalities with supremum operators on the cone of monotone functions. In particular, the study of the case with non-decreasing functions was initiated in this paper.

In paper F we focus only on the much less studied problem, namely to characterize Hardy-type inequalities on the cone of non-decreasing functions. A new reduction method is used in a crucial way. Some complete charac-terizations for all studied cases are discussed and proved. The investigations initiated in paper C are here developed to a more general theory, which cov-ers all studied operators. The obtained results are used to derive some new bilinear Hardy-type inequalities.

We study the problem of characterizing the weighted inequalities on the Lebesgue cones of monotone functions on the semiaxis for a class of quasilinear integral operators.

We solve the characterization problem of L^p _v -L^r _p weighted inequalities on Lebesgue cones of monotone functions on the half-axis for quasilinear integral operators of iterated type with Oinarov's kernels.

We study the problem of characterizing weighted inequalities on Lebesgue cones of monotone functions on the half-axis for one class of quasilinear integral operators.