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. A475, 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.

在本文中，我们继续研究给出两个权重 Hardy 不等式的权重特征，以保持具有极性分解的一般度量度量空间。由于在这些空间上可能没有可微结构，因此本着哈代原始不等式的精神，不等式以积分形式给出。这是我们论文 (Ruzhansky & Verma 2018. Proc. R. Soc. A 475 , 20180310 (doi:10.1098/rspa.2018.0310)) 的延续，其中我们处理了p  ≤  q的情况。这里考虑剩余范围p  >  q，即 0 <  q  <  p , 1 <  p < ∞。我们给出了所获得结果的几个例子，找到了齐次群、双曲空间和更一般的 Cartan-Hadamard 流形上的积分 Hardy 不等式的权重条件。与本文的第一部分一样，我们不需要对指标施加加倍条件。