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

Abstract

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We consider a generalized Hardy operator T f ( x ) = ϕ ( x ) ʃ 0 x ψ f v . 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 = s u p R > 0 ∥ ϕ χ ( R , ∞ ) ∥ Y ∥ ψ χ ( 0 , R ) ∥ X ' < ∞ 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).

How to cite

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Edmunds, 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 -

References

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