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 Hardy operator, Hardy-Steklov operator, compactness, boundedness, star-shaped regions

MSC Classifications:

46E35, 26D10 show english descriptions Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
Inequalities involving derivatives and differential and integral operators 46E35 - Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
26D10 - Inequalities involving derivatives and differential and integral operators

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