Weighted Hardy-type inequalities on the cones of monotone and quasi-concave functions

This PhD thesis deals with weighted Hardy-type inequalities restricted to cones of monotone functions and quasi-monotone functions.The thesis consists of four papers (papers A, B, C and D) and an introduction, which gives an overview of this specific field of functional analysis and also serves to put these papers into a more general frame. The papers A and B are devoted to characterizing some weighted Hardy-type inequalities on the cones of monotone functions, while in the papers C and D we solve the similar problems for the cones of quasi-concave and $\psi-$quasi-concave functions.In paper A some two-sided inequalities for Hardy operators on the cones of monotone functions are proved for the full range of parameter $1<p<\infty.$ As an application we obtain a new characterization of the discrete Hardy inequality for the most difficult case of parameters, namely when $0<q<p\leq 1.$ The equivalences we proved in paper A are used in paper B to obtain necessary and sufficient conditions for some other Hardy-type inequalities on the cones of monotone functions. In particular, we give a complete description for inequalities with Volterra integral operators involving Oinarov's kernel for parameters $0<p<\infty,\,\,1\leq q<\infty.$ Paper C is motivated by the well-known characterization problem of the boundedness of the classical Hardy-Littlewood maximal operator in weighted Lorentz $\Gamma-$spaces. Here we introduce the classes of quasi-concave and $\psi-$quasi-concave functions and derive criteria for some Hardy-type inequalities to hold on the cone of $\psi-$quasi-concave functions for the range of parameters $0<p<\infty, ,\,1\leq q<\infty.$ For the rest of the range we come up with the sufficient conditions. In paper D we extend the method of characterization we have used in paper C for the cone of quasi-concave functions to a cone of $\psi-$quasi-concave functions to obtain yet another characterization for one of the inequalities to hold on this cone. As an application we obtain some new results concerning mapping properties of the Hilbert transform and Riesz potentials in weighted Lorentz spaces.Godkänd; 2012; 20120502 (olgpop); DISPUTATION Ämnesområde: Matematik/Mathematics Opponent: Professor Mikhail Goldman, Peoples Friendship University Moscow, Russia, Ordförande: Professor Lars-Erik Persson, Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet Tid: Fredag den 15 juni 2012, kl 10.00 Plats: D2214, Luleå tekniska universite