an: Zbl 1193.26013
au: Stepanov, V.D.; Ushakova, E.P.
ti: On boundedness of a certain class of Hardy-Steklov type operators in
    Lebesgue spaces.
la: EN
so: Banach J. Math. Anal. 4, No. 1, 28-52, electronic only (2010).
py: 2010
dt: J
cc: *26D10 26D15 26D07
ut: integral operator; Hardy-Steklov type operator; boundedness; weight
    function; Oinarov's condition
ab: 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 $\mathcal{O}_a$.
rv: Mehdi Hassani (Zanjan)