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1.

Persson, Lars-Erik

et al.

Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Narvik University College.

Shambilova, Guldarya E.

Department of Mathematical Analysis and Function Theory, Peoples Friendship University, Moscow.

Stepanov, Vladimir D

Department of Mathematical Analysis and Function Theory, Peoples Friendship University of Russia; Steklov Mathematical Institute, Russian Academy of Sciences.

Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. UiT, The Arctic University of Norway, Narvik, Norway.

Shambilova, Guldarya E.

Department of Mathematics, Financial University under the Government of the Russian Federation.

Stepanov, Vladimir D.

Department of Nonlinear Analysis and Optimization, Peoples’ Friendship University of Russia.

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.

7.

Stepanov, V.D.

et al.

Peoples’ Friendship University of Russia Steklov Institute of Mathematics, Moscow, Russia.

Shambilova, Guldarya E.

Financial University Under the Government of the Russian Federation, Moscow, Russia.

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.