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In this note we continue giving the characterisation of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality. This is a continuation of our paper [M. Ruzhansky and D. Verma. Hardy inequalities on metric measure spaces, Proc. R. Soc. A., 475(2223):20180310, 2018] where we treated the case $p\leq q$. Here the remaining range $p>q$ is considered, namely, $0, Comment: 18 pages; this is the second part to the paper arXiv:1806.03728. Final version, to appear in Proc. Royal Soc. A

In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality. We give examples obtaining new weighted Hardy inequalities on $\mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds., Comment: 18 pages; typos have been corrected

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.

Boundedness of the Hardy operator ( H f ) ( x ) = ∫ 0 x f ( t ) d t and its adjoint ( H ∗ f ) ( x ) = ∫ x ∞ f ( t ) d t is characterized between Banach function spaces X q and L p . By applying a limiting procedure, corresponding boundedness of the geometric mean operator ( G f ) ( x ) = exp ( 1 x ∫ 0 x ln f ( t ) d t ) is also derived.