Geodesic
On some structural properties of Banach function spaces and boundedness of certain integral operators
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 791-805.

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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 \[ K f(x)=\varphi (x) \int _0^x k(x,y)f(y)\psi (y)\mathrm{d}y \] 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.
Classification : 42B20, 42B25, 45P05, 46E30, 47G10
Keywords: Banach function space; uniformly upper; uniformly lower $\ell $-estimate; Hardy type operator 
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     title = {On some structural properties of {Banach} function spaces and boundedness of certain integral operators},
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Kopaliani, T. S. On some structural properties of Banach function spaces and boundedness of certain integral operators. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 791-805. http://geodesic.mathdoc.fr/item/CMJ_2004__54_3_a20/