Sharp bounds for generalized Hardy operator on product space

Abstract

We characterize a sufficient and necessary condition which ensures that the generalized Hardy operator U ψ f( x )= \begin{document} $\int\limits_0^1 {} \cdots \int\limits_0^1 {} $ \end{document} f( x 1 t 1 ,…, x n t n )ψ( t 1 ,…, t n ) d t 1 … d t n is bounded on RMO( \begin{document} $\mathbb{R}$ \end{document} n ).The condition deeply depends on the nonnegative function ψ defined on [0,1]×…×[0,1].Furthermore, the corresponding operator norm is worked out.In addition, we also extend the results to the high-dimensional product space.