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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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References:
[1] C. Bennett and R. Sharpley: Interpolation of Operators. Acad. Press, Boston, 1988. · Zbl 0647.46057
[2] J. Lindenstrauss and L. Tzafriri: Classical Banach Spaces. II. Function Spaces. Springer-Verlag, 1979. · Zbl 0403.46022
[3] A. V. Bukhvalov, V. B. Korotkov, A. G. Kusraev, S. S. Kutateladze and B. M. Makarov: Vector Lattices and Integral Operators. Nauka, Novosibirsk, 1992. (In Russian.) · Zbl 0752.46001
[4] J. Musielak: Orlicz Spaces and Modular Spaces. Lecture Notes in Math. 1034. Springer-Verlag, Berlin-Heidelberg-New York, 1983. · Zbl 0557.46020
[5] V. D. Stepanov: Nonlinear Analysis. Function Spaces and Applications 5. Olympia Press, 1994, pp. 139-176.
[6] E. N. Lomakina and V. D. Stepanov: On Hardy type operators in Banach function spaces on half-line. Dokl. Akad. Nauk 359 (1998), 21-23. (In Russian.) · Zbl 0958.47022
[7] P. Oinarov: Two-side estimates of the norm of some classes of integral operators. Trudy Mat. Inst. Steklov. 204 (1993), 240-250. (In Russian.)
[8] A. V. Bukhvalov: Generalization of Kolmogorov-Nagumo’s theorem on tensor product. Kachestv. pribl. metod. issledov. operator. uravnen. 4 (1979), 48-65. (In Russian.)
[9] E. I. Berezhnoi: Sharp estimates for operators on cones in ideal spaces. Trudy Mat. Inst. Steklov. 204 (1993), 3-36. (In Russian.)
[10] E. I. Berezhnoi: Two-weighted estimations for the Hardy?Littlwood maximal function in ideal Banach spaces. Proc. Amer. Math. Soc. 127 (1999), 79-87. · Zbl 0918.42011
[11] Q. Lai: Weighted modular inequalities for Hardy type operators. Proc. London Math. Soc. 79 (1999), 649-672. · Zbl 1030.46030
[12] I. I. Sharafutdinov: On the basisity of the Haar system in L p(t) ([0; 1]) spaces. Mat. Sbornik 130 (1986), 275-283. (In Russian.)
[13] I. I. Sharafutdinov: The topology of the space L p(t) ([0; 1]). Mat. Zametki 26 (1976), 613-632. (In Russian.)
[14] O. Kov??ik and J. R?kosn?k: On spaces L p(x) and W k;p(x). Czechoslovak Math.J. 41 (1991), 592-618.
[15] H. H. Schefer: Banach Lattices and Positive Operators. Springer-Verlag, Berlin-Heidel-berg-New York, 1974.
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