Hardy-type inequalities on the cones of monotone functions

Abstract

This Licentiate thesis deals with Hardy-type inequalities restricted to cones of monotone functions. The thesis consists of two papers (paper A and paper B) and an introduction which gives an overview to this specific field of functional analysis and also serves to put the papers into a more general frame. We deal with positive σ-finite Borel measures on R + := [0, ∞) and the class M ↓ (M ↑) consisting of all non-increasing (non-decreasing) Borel functions f : R + → [0, +∞]. In paper A some two-sided inequalities for Hardy operators on the cones of monotone functions are proved. The idea to study such equivalences follows from the Hardy inequality

Weight boundedness of a class of quasilinear operators on the cone of monotone functions

“…In the case r = q, these inequalities were studied by many authors (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references in survey [11]), but the case r ≠ q has not been investi gated; handling it has required a fresh idea borrowed from [14,15].…”

Hardy's operator and normability of generalized Lorentz-Marcinkiewicz spaces, with sharp or weakly sharp constant estimation

“…See also [1], [9], [41] where are obtained alike result without constants computation. In the theses of L.Arendarenko [2] and O.Popova [52] there is a comprehensive review about this problem and are offered some new results.…”