## 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)
Full Text: