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The generalized Hardy operator with kernel and variable integral limits in Banach function spaces. (English) Zbl 0947.47020
The authors consider the integral operator $Kf(x):=v(x)\int_{a(x)}^{b(x)} k(x,y) u(y) f(y)dy$ for $$x>0$$, where $$a$$ and $$b$$ are nondecreasing functions, $$u$$ and $$v$$ are non-negative and finite functions, and $$k(x,y)\geq 0$$ is nondecreasing in $$x$$, nondecreasing in $$y$$ and $$k(x,z)\leq D[k(x,b(y))+k(y,z)]$$ for $$y\leq x$$ and $$a(x)\leq z \leq b(y)$$. It is shown that the integral operator $$K: X\rightarrow Y$$, where $$X$$ and $$Y$$ are Banach function spaces with $$l$$-condition is bounded if and only if $\sup_{x\leq y, a(y)\leq b(x)}\|\chi_{(x, y)}(\cdot)v(\cdot)k(\cdot, b(x))\|_Y\|\chi_{a(y), b(x)}u\|_{X'}+$ $\sup_{x\leq y, a(y)\leq b(x)}\|\chi_{(x, y)} v\|_Y\|\chi_{(a(y), b(x))}(\cdot)K(x,\cdot)u(\cdot)\|_{X'}<\infty .$

##### MSC:
 47B34 Kernel operators 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47B38 Linear operators on function spaces (general)
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