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 R n , on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds.

In this paper, we continue our investigations giving the characterization 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 (Ruzhansky & Verma 2018. Proc. R. Soc. A 475 , 20180310 ( doi:10.1098/rspa.2018.0310 )) where we treated the case p  ≤  q . Here the remaining range p  >  q is considered, namely, 0 <  q  <  p , 1 <  p  < ∞. We give several examples of the obtained results, finding conditions on the weights for integral Hardy inequalities on homogeneous groups, as well as on hyperbolic spaces and on more general Cartan–Hadamard manifolds. As in the first part of this paper, we do not need to impose doubling conditions on the metric.

Boundedness of the Hardy operatoris characterized between Banach function spaces X q and L p . By applying a limiting procedure, corresponding boundedness of the geometric mean operator (Gf )(x) = exp( 1 x x 0 ln f (t) dt) is also derived.