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

We consider a generalized Hardy operator ${\displaystyle Tf\left(x\right)=\varphi \left(x\right){ʃ}_0^x\psi fv}$. 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 ${\displaystyle ℬ={\supset }_{R>0}\parallel \varphi {\chi }_{(R,\infty )}{\parallel }_Y\parallel \psi {\chi }_{(0,R)}{\parallel }_{X\prime }<\infty }$ 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).