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Zbl 1193.26013
Stepanov, V.D.; Ushakova, E.P.
On boundedness of a certain class of Hardy-Steklov type operators in Lebesgue spaces. (English)
[J] Banach J. Math. Anal. 4, No. 1, 28-52, electronic only (2010). ISSN 1735-8787/e

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 $\mathcal{O}_a$.
[Mehdi Hassani (Zanjan)]

MSC 2000:: *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

Keywords: integral operator; Hardy-Steklov type operator; boundedness; weight function; Oinarov's condition

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