The hardy operator and the gap between L∞ and BMO

We study boundedness and compactness properties of the Hardy integral operator Tf(x) = integral(A)(x) f from a weighted Banach function space X(v) into L-infinity and BMO. We give a new simple characterization of compactness of T from X(v) into BMO. We construct examples of spaces X(v) such that T(X(v)) is (a) bounded in L-infinity but not compact in BMO; (b) compact in BMO but not bounded in L-infinity; (c) bounded in BMO but neither bounded in L-infinity nor compact in BMO; (d) bounded in L-infinity, compact in BMO and yet not compact in L-infinity. In order to obtain the last of the counterexamples we construct a new weighted Banach function space.