Geodesic
Compactness criterion for fractional integration operator of infinitesimal order
Ufa mathematical journal, Tome 5 (2013) no. 1, pp. 3-10

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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 $1$ and $v(x)=x^{-\gamma}$, $\gamma>0$, where $L_{q,u}$ is the set of all measurable on $(0, \infty)$ functions $f$ with finite norm $\|uf\|_{q}$.
Keywords: compactness, fractional integration operator, Riemann–Liouville operator, singular operator, adjoint operator, Holder inequality, weighted inequalities. 
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A. M. Abylayeva; A. O. Baiarystanov. Compactness criterion for fractional integration operator of infinitesimal order. Ufa mathematical journal, Tome 5 (2013) no. 1, pp. 3-10. http://geodesic.mathdoc.fr/item/UFA_2013_5_1_a0/