## Banach Journal of Mathematical Analysis

### On the boundedness and compactness of a certain integral operator

S. M. Farsani

#### 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.

#### Article information

Source
Banach J. Math. Anal., Volume 7, Number 2 (2013), 86 -102 .

Dates
First available in Project Euclid: 20 March 2013

https://projecteuclid.org/euclid.bjma/1363784225

Digital Object Identifier
doi:10.15352/bjma/1363784225

Mathematical Reviews number (MathSciNet)
MR3039941

Zentralblatt MATH identifier
1270.26019

Subjects
Primary: 26D10
Secondary: 47G10 26D15 26D07

#### Citation

Farsani , S. M. On the boundedness and compactness of a certain integral operator. Banach J. Math. Anal. 7 (2013), no. 2, 86 --102. doi:10.15352/bjma/1363784225. https://projecteuclid.org/euclid.bjma/1363784225

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