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

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E. Edmunds D., Gurka P., Pick L.

Studia Mathematica

1994

tom 109

nr 1

73-90

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 $ℬ = sup_{R>0} ∥ϕχ_{(R,∞)}∥_{Y}∥ψχ_{(0,R)}∥_{X'} < ∞$ 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).

Ograniczanie wyników

1 Studia Mathematica

1 E. Edmunds D.

1 Gurka P.

1 Pick L.

1 1994

Wyniki wyszukiwania

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