On some structural properties of Banach function spaces and boundedness of certain integral operators

T. S. Kopaliani

Czechoslovak Mathematical Journal (2004)

  • Volume: 54, Issue: 3, page 791-805
  • ISSN: 0011-4642

Abstract

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In this paper the notions of uniformly upper and uniformly lower ℓ -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 ℓ -estimate and uniformly an upper ℓ -estimate, respectively. The integral operator from X into Y of the form K f ( x ) = ϕ ( x ) ∫ 0 x k ( x , y ) f ( y ) ψ ( y ) d y is studied, where k , ϕ , ψ 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.

How to cite

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Kopaliani, T. S.. "On some structural properties of Banach function spaces and boundedness of certain integral operators." Czechoslovak Mathematical Journal 54.3 (2004): 791-805. <http://eudml.org/doc/30901>.

@article{Kopaliani2004,
abstract = {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.},
author = {Kopaliani, T. S.},
journal = {Czechoslovak Mathematical Journal},
keywords = {Banach function space; uniformly upper; uniformly lower $\ell $-estimate; Hardy type operator; Banach function space; uniformly upper and uniformly lower -estimate; Hardy type operator},
language = {eng},
number = {3},
pages = {791-805},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On some structural properties of Banach function spaces and boundedness of certain integral operators},
url = {http://eudml.org/doc/30901},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Kopaliani, T. S.
TI - On some structural properties of Banach function spaces and boundedness of certain integral operators
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 3
SP - 791
EP - 805
AB - 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.
LA - eng
KW - Banach function space; uniformly upper; uniformly lower $\ell $-estimate; Hardy type operator; Banach function space; uniformly upper and uniformly lower -estimate; Hardy type operator
UR - http://eudml.org/doc/30901
ER -

References

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  11. Weighted modular inequalities for Hardy type operators, Proc. London Math. Soc. 79 (1999), 649–672. (1999) Zbl1030.46030MR1710168
  12. On the basisity of the Haar system in L p ( t ) ( [ 0 , 1 ] ) spaces, Mat. Sbornik 130 (1986), 275–283. (Russian) (1986) 
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