Compactness criterion for fractional integration operator of infinitesimal order

We obtain necessary and sufficient conditions of compactness for the operator Kf(x) = x ∫ 0 ln x x− s f(s) s ds from Lp,v in Lq,u as 1 0, where Lq,u is the set of all measurable on (0,∞) functions f with finite norm ‖uf‖q.