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Compactness of Hardy-type integral operators in weighted Banach function spaces. (English) Zbl 0821.46036

Summary: We consider a generalized Hardy operator \(Tf(x)= \phi(x) \int^ x_ 0 \psi fv\). 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 \({\mathcal B}= \sup_{R> 0} \| \phi\chi_{(R,\infty)}\|_ Y\| \psi\chi_{(0, R)}\|_{X'}< \infty\) be satisfied. We say that \((X, Y)\) belongs to the category \({\mathcal M}(T)\) if this Muckenhoupt condition is also sufficient. We prove a general criterion for compactness of \(T\) from \(X\) to \(Y\) when \((X, Y)\in {\mathcal M}(T)\) 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 \({\mathcal M}(T)\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B38 Linear operators on function spaces (general)
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