Article
Ufa Mathematical Journal
Volume 5, Number 1, pp. 3-10
Compactness criterion for fractional integration operator of infinitesimal order
Abylayeva А.М., Baiarystanov А.О.
DOI:10.13108/2013-5-1-3
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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 $L_{q,u}$ as $ 1 < p\leq q < \infty $ 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 $\|uf\|_{q}$.