Two-Dimensional Mean Inequalities in Certain Banach Function Spaces

Abstract

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.