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On some structural properties of Banach function spaces and boundedness of certain integral operators. (English) Zbl 1080.47040
Summary: In this paper, the notions of uniformly upper and uniformly lower $$\ell$$-estimates for Banach function spaces are introduced. Further, the pair $$(X,Y)$$ of Banach function spaces is characterized, where $$X$$ and $$Y$$ satisfy uniformly a lower $$\ell$$-estimate and uniformly an upper $$\ell$$-estimate, respectively. The integral operator from $$X$$ into $$Y$$ of the form $K f(x)=\varphi (x) \int _0^x k(x,y)f(y)\psi (y)\,\text dy$ is studied, where $$k$$, $$\varphi$$, $$\psi$$ are prescribed functions under some local integrability conditions, the kernel $$k$$ is non-negative and is assumed to satisfy certain additional conditions, notably one of monotone type.

##### MSC:
 47G10 Integral operators 45P05 Integral operators 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory
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