| Title:
|
On some structural properties of Banach function spaces and boundedness of certain integral operators (English) |
| Author:
|
Kopaliani, T. S. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
3 |
| Year:
|
2004 |
| Pages:
|
791-805 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 \[ K f(x)=\varphi (x) \int _0^x k(x,y)f(y)\psi (y)\mathrm{d}y \] 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. (English) |
| Keyword:
|
Banach function space |
| Keyword:
|
uniformly upper |
| Keyword:
|
uniformly lower $\ell $-estimate |
| Keyword:
|
Hardy type operator |
| MSC:
|
42B20 |
| MSC:
|
42B25 |
| MSC:
|
45P05 |
| MSC:
|
46E30 |
| MSC:
|
47G10 |
| idZBL:
|
Zbl 1080.47040 |
| idMR:
|
MR2086735 |
| . |
| Date available:
|
2009-09-24T11:17:44Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127930 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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[4] J. Musielak: Orlicz Spaces and Modular Spaces. Lecture Notes in Math. 1034.Springer-Verlag, Berlin-Heidelberg-New York, 1983. MR 0724434 |
| Reference:
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| Reference:
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[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. (Russian) MR 1668395 |
| Reference:
|
[7] P. Oinarov: Two-side estimates of the norm of some classes of integral operators.Trudy Mat. Inst. Steklov. 204 (1993), 240–250. (Russian) MR 1320028 |
| Reference:
|
[8] A. V. Bukhvalov: Generalization of Kolmogorov-Nagumo’s theorem on tensor product.Kachestv. pribl. metod. issledov. operator. uravnen. 4 (1979), 48–65. (Russian) |
| Reference:
|
[9] E. I. Berezhnoi: Sharp estimates for operators on cones in ideal spaces.Trudy Mat. Inst. Steklov. 204 (1993), 3–36. (Russian) MR 1320016 |
| Reference:
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[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. MR 1622773, 10.1090/S0002-9939-99-04998-9 |
| Reference:
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[11] Q. Lai: Weighted modular inequalities for Hardy type operators.Proc. London Math. Soc. 79 (1999), 649–672. Zbl 1030.46030, MR 1710168 |
| Reference:
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[12] I. I. Sharafutdinov: On the basisity of the Haar system in $L^{p(t)}([0,1])$ spaces.Mat. Sbornik 130 (1986), 275–283. (Russian) |
| Reference:
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[13] I. I. Sharafutdinov: The topology of the space $L^{p(t)}([0,1])$.Mat. Zametki 26 (1976), 613–632. (Russian) |
| Reference:
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[14] O. Kováčik and J. Rákosník: On spaces $L^{p(x)}$ and $W^{k,p(x)}$.Czechoslovak Math. J. 41 (1991), 592–618. MR 1134951 |
| Reference:
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[15] H. H. Schefer: Banach Lattices and Positive Operators.Springer-Verlag, Berlin-Heidelberg-New York, 1974. |
| . |
