On the boundedness and compactness of a certain integral operator

Abstract

Let $\alpha \in (0, \infty)$ and $\beta \in (1, \infty)$. In the present work, the necessary and sufficient conditions for the boundedness and compactness of the integral operator of the form \begin{equation*} L_{\alpha, \beta} f(x):=v(x)\int_0^x \frac{\ln^{\beta-1}(\frac{x}{y})f(y)u(y)dy}{(x-y)^{1-\alpha}} ,\,\,\,\, x>0, \end{equation*} from $L^p\to L^q$, with locally integrable non-negative weight functions $u$ and $v$, in the case $p,q \in (0, \infty), p>\max(1/{\alpha},1),$ provided $u$ is non-increasing on $\mathbb{R}^+:=[0,\infty)$ are found.