@article{1193.26013,
author="Stepanov, V.D. and Ushakova, E.P.",
title="{On boundedness of a certain class of Hardy-Steklov type operators in
    Lebesgue spaces.}",
language="English",
journal="Banach J. Math. Anal. ",
volume="4",
number="1",
pages="28-52",
year="2010",
abstract="{Suppose that $w$ and $v$ are locally integrable non-negative weight
    functions. The paper under review studies $L_p$-$L_q$ boundedness of the
    Hardy-Steklov type operator $\mathcal{K}$ defined by $$
    \mathcal{K}f(x)=w(x)\int_{a(x)}^{b(x)}k(x,y)f(y)v(y)dy, $$ where the border
    functions $a(x)$ and $b(x)$ are differentiable and strictly increasing on
    $(0,\infty)$, $a(0)=b(0)=0$, $a(x)<b(x)$ for $x\in (0,\infty)$,
    $a(\infty)=b(\infty)=\infty$, and the kernel $k(x,y)$ is strictly positive
    on the set $\{(x,y):x>0,a(x)<y<b(x)\}$ and satisfies at least one of two
    generalized Oinarov's conditions $\mathcal{O}_b$ and $\mathcal{O}_a$.}",
reviewer="{Mehdi Hassani (Zanjan)}",
keywords="{integral operator; Hardy-Steklov type operator; boundedness; weight
    function; Oinarov's condition}",
classmath="{*26D10 (Inequalities involving derivatives, diff. and integral
    operators)
26D15 (Inequalities for sums, series and integrals of real functions)
26D07 (Inequalities involving other types of real functions)
}",
}