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# Compactness of Hardy-Type Operators over Star-Shaped Regions in $\mathbb{R}^N$

Published:2004-12-01
Printed: Dec 2004
• Pankaj Jain
• Pawan K. Jain
• Babita Gupta
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## Abstract

We study a compactness property of the operators between weighted Lebesgue spaces that average a function over certain domains involving a star-shaped region. The cases covered are (i) when the average is taken over a difference of two dilations of a star-shaped region in $\RR^N$, and (ii) when the average is taken over all dilations of star-shaped regions in $\RR^N$. These cases include, respectively, the average over annuli and the average over balls centered at origin.
 Keywords: Hardy operator, Hardy-Steklov operator, compactness, boundedness, star-shaped regions
 MSC Classifications: 46E35 - Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems 26D10 - Inequalities involving derivatives and differential and integral operators