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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.

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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