Ruzhansky Michael, Verma Daulti
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan, 281, Building S8, B, 9000 Ghent, Belgium.
School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK.
Proc Math Phys Eng Sci. 2021 Jun;477(2250):20210136. doi: 10.1098/rspa.2021.0136. Epub 2021 Jun 9.
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. , 20180310 (doi:10.1098/rspa.2018.0310)) where we treated the case ≤ . Here the remaining range > is considered, namely, 0 < < , 1 < < ∞. 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，我们继续进行研究，给出在具有极分解的一般度量测度空间上使双权Hardy不等式成立的权函数的特征描述。由于此类空间可能不存在可微结构，所以这些不等式是按照Hardy原始不等式的精神以积分形式给出的。这是我们论文（Ruzhansky & Verma 2018. ，20180310 (doi:10.1098/rspa.2018.0310)）的延续，在那篇论文中我们处理了 ≤ 的情况。这里考虑剩余的范围 > ，即0 < < ，1 < < ∞。我们给出了几个所得结果的例子，找到了齐性群、双曲空间以及更一般的Cartan - Hadamard流形上积分Hardy不等式的权函数条件。如同本文第一部分一样，我们无需对度量施加加倍条件。
