Compactness criterion for fractional integration operator of infinitesimal order

Abstract

We obtain necessary and sufficient conditions of compactness for the operator $$Kf(x)=\int\limits_{0}^{x}\ln\frac{x}{x-s}\frac{f(s)}{s}ds$$ from $L_{p,v}$ in $L_{q,u}$ at $1

0$, where $L_{q,u}$ is the set of all measurable on $(0, \infty)$ functions $f$ with finite norm $\|uf\|_{q}$.