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

A. A. Kalybay ; R. Oinarov

Izvestiya. Mathematics , Tome 83 (2019) no. 2, pp. 251-272

Résumé

We consider weighted bounds for quasilinear integral operators of the form $$ \mathcal{K}^+f(x)=\biggl(\int_{0}^{x}\biggl|w(t)\int_{t}^{x} K(s,t)f(s)\,ds\biggr|^{r}\,dt\biggr)^{{1}/{r}} $$ 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 $0$, 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 $1\leqslant p\leqslant q\infty$ and $0$, $p\geqslant 1$, and also on the cones of non-negative non-increasing and non-negative non-decreasing functions for $0$ and $1\leqslant p\infty$. Here it is assumed only that $K{(\,\cdot\,,\cdot\,)}\geqslant 0$. However, the criteria we obtain involve the norm of a linear integral operator from $L_{p,v}$ to $L_{r,w}$ with kernel $K{(\,\cdot\,,\cdot\,)}$.

Détail

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