# Compactness of Hardy-type integral operators in weighted Banach function spaces

David Edmunds; Petr Gurka; Luboš Pick

Studia Mathematica (1994)

- Volume: 109, Issue: 1, page 73-90
- ISSN: 0039-3223

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topEdmunds, David, Gurka, Petr, and Pick, Luboš. "Compactness of Hardy-type integral operators in weighted Banach function spaces." Studia Mathematica 109.1 (1994): 73-90. <http://eudml.org/doc/216062>.

@article{Edmunds1994,

abstract = {We consider a generalized Hardy operator $Tf(x) = ϕ(x) ʃ_\{0\}^\{x\} ψfv$. For T to be bounded from a weighted Banach function space (X,v) into another, (Y,w), it is always necessary that the Muckenhoupt-type condition $ℬ = sup_\{R>0\} ∥ϕχ_\{(R,∞)\}∥_\{Y\}∥ψχ_\{(0,R)\}∥_\{X^\{\prime \}\} < ∞$ be satisfied. We say that (X,Y) belongs to the category M(T) if this Muckenhoupt condition is also sufficient. We prove a general criterion for compactness of T from X to Y when (X,Y) ∈ M(T) and give an estimate for the distance of T from the finite rank operators. We apply the results to Lorentz spaces and characterize pairs of Lorentz spaces which fall into M (T).},

author = {Edmunds, David, Gurka, Petr, Pick, Luboš},

journal = {Studia Mathematica},

keywords = {weighted Banach function space; Hardy-type operator; compact operator; Lorentz space; generalized Hardy operator; Muckenhoupt condition; compactness; finite rank operators; Lorentz spaces},

language = {eng},

number = {1},

pages = {73-90},

title = {Compactness of Hardy-type integral operators in weighted Banach function spaces},

url = {http://eudml.org/doc/216062},

volume = {109},

year = {1994},

}

TY - JOUR

AU - Edmunds, David

AU - Gurka, Petr

AU - Pick, Luboš

TI - Compactness of Hardy-type integral operators in weighted Banach function spaces

JO - Studia Mathematica

PY - 1994

VL - 109

IS - 1

SP - 73

EP - 90

AB - We consider a generalized Hardy operator $Tf(x) = ϕ(x) ʃ_{0}^{x} ψfv$. For T to be bounded from a weighted Banach function space (X,v) into another, (Y,w), it is always necessary that the Muckenhoupt-type condition $ℬ = sup_{R>0} ∥ϕχ_{(R,∞)}∥_{Y}∥ψχ_{(0,R)}∥_{X^{\prime }} < ∞$ be satisfied. We say that (X,Y) belongs to the category M(T) if this Muckenhoupt condition is also sufficient. We prove a general criterion for compactness of T from X to Y when (X,Y) ∈ M(T) and give an estimate for the distance of T from the finite rank operators. We apply the results to Lorentz spaces and characterize pairs of Lorentz spaces which fall into M (T).

LA - eng

KW - weighted Banach function space; Hardy-type operator; compact operator; Lorentz space; generalized Hardy operator; Muckenhoupt condition; compactness; finite rank operators; Lorentz spaces

UR - http://eudml.org/doc/216062

ER -

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## Citations in EuDML Documents

top- W. Evans, D. Harris, J. Lang, Two-sided estimates for the approximation numbers of Hardy-type operators in ${L}^{\infty}$ and L¹
- Edmunds, David E., Recent developments concerning entropy and approximation numbers
- Stepanov, Vladimir D., Weighted norm inequalities for integral operators and related topics

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