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2007/2008 Two-Dimensional Mean Inequalities in Certain Banach Function Spaces
Pankaj Jain, Daulti Verma
Real Anal. Exchange 33(1): 127-144 (2007/2008).


Weight characterization is obtained for the $L^p$-$X^q$ boundedness of the two-dimensional Hardy operator $(H_2f)(x_1,x_2)=\int_0^{x_1}\int_0^{x_2}f(t_1,t_2)\,dt_1\,dt_2$. By using a limiting procedure as well as by a direct method, the corresponding boundedness of the two-dimensional geometric mean operator $(G_2f)(x_1,x_2)=\exp\bigg(\dfrac{1}{x_1x_2}\int_0^{x_1}\int_0^{x_2}\ln f(t_1,t_2)\,dt_1\,dt_2\bigg)$ is obtained.


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Pankaj Jain. Daulti Verma. "Two-Dimensional Mean Inequalities in Certain Banach Function Spaces." Real Anal. Exchange 33 (1) 127 - 144, 2007/2008.


Published: 2007/2008
First available in Project Euclid: 28 April 2008

zbMATH: 1152.26014
MathSciNet: MR2402868

Primary: 26D10 , 26D15

Keywords: Banach function space , ‎geometric mean operator , Hardy inequality , Hardy operator , two dimensional inequality

Rights: Copyright © 2007 Michigan State University Press

Vol.33 • No. 1 • 2007/2008
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