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Izv. RAN. Ser. Mat., 2019, Volume 83, Issue 2, Pages 61–82 (Mi izv8613)  

Bounds for a class of quasilinear integral operators on the set of non-negative and non-negative monotone functions

A. A. Kalybaya, R. Oinarovb

a Университет КИМЭП, г. Алматы, Казахстан
b L. N. Gumilev Eurasian National University, Astana

Abstract: We consider weighted bounds for quasilinear integral operators of the form
$$ \mathcal{K}^+f(x)=(\int_{0}^{x}|w(t)\int_{t}^{x} K(s,t)f(s) ds|^{r} dt)^{{1}/{r}} $$
from $L_{p,v}$ to $L_{q,u}$ on the set on non-negative and non-negative monotone functions $f$, where $u$, $v$ and $w$ are weight functions. Under the assumption that $0<r<\infty$, we obtain necessary and sufficient conditions for the validity of these bounds on the set of non-negative functions for the values of the parameters satisfying the conditions $1\leq p\leq q<\infty$ and $0<q<p<\infty$, $p\geq 1$, and also on the cones of non-negative non-increasing and non-negative non-decreasing functions for $0<q<\infty$ and $1\leq p<\infty$. Here it is assumed only that $K{( \cdot ,\cdot )}\geq 0$. However, the criteria we obtain involve the norm of a linear integral operator from $L_{p,v}$ to $L_{r,w}$ with kernel $K{( \cdot ,\cdot )}$.

Keywords: integral operator, inequality of Hardy type, weight function, kernel, monotone function.

Funding Agency Grant Number
Ministry of Education and Science of the Republic of Kazakhstan AP05130975
This work was financially supported by the Ministry of Education and Science of the Republic of Kazakhstan, grant no. AP05130975 in the area ‘Scientific research in the natural sciences’.

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/im8613

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English version:
Izvestiya: Mathematics, 2019, 83:2, 251–272

Bibliographic databases:

UDC: 517.51
MSC: 26D10, 47B38
Received: 07.10.2016
Revised: 25.03.2017

Citation: A. A. Kalybay, R. Oinarov, “Bounds for a class of quasilinear integral operators on the set of non-negative and non-negative monotone functions”, Izv. RAN. Ser. Mat., 83:2 (2019), 61–82; Izv. Math., 83:2 (2019), 251–272

Citation in format AMSBIB
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