Weighted norm inequalities for positive operators restricted on the cone of λ-quasiconcave functions

0522215 - MÚ 2021 RIV GB eng J - Journal Article
Gogatishvili, Amiran - Neves, J. S.
Weighted norm inequalities for positive operators restricted on the cone of λ-quasiconcave functions.
Proceedings of the Royal Society of Edinburgh. A - Mathematics. Roč. 150, č. 1 (2020), s. 17-39. ISSN 0308-2105. E-ISSN 1473-7124
R&D Projects: GA ČR GA13-14743S
Institutional support: RVO:67985840
Keywords : quasilinear operator * integral inequality * Lebesgue space * Hardy operator * quasiconcave functions * monotone functions
OECD category: Pure mathematics
Impact factor: 1.319, year: 2020
Method of publishing: Open access
https://doi.org/10.1017/prm.2018.85

Let ρ be a monotone quasinorm de_ned on M^+, the set of all non-negative measurable functions on [0,1): Let T be a monotone quasilinear operator on M^+. We show that the following inequality restricted on the cone of λ-quasiconcave functions ρ(f)≤C(∫_0^∞ f^p v)^(1/p), where 1≤p≤∞ and v is a weighted function, is equivalent to slightly different inequalities consider for all non-negative measurable functions. The case 0 < p < 1 is also studied for quasinorms and operators with additional properties. These results in turn enables us to establish necessary and sufficient conditions on the weights (u, v,w) for which the three weighted Hardy-type inequalityholds for all ρ-quasiconcave functions and all 0 < p,q ≤∞.
Permanent Link: http://hdl.handle.net/11104/0306710