Abstract
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.
Highlights
After the Hardy inequality was proved by Hardy in [1], a large amount of literature is available on this inequality
In our previous paper [21], for the case 1 < p ≤ q < ∞, we characterized the weights u and v for the Hardy inequalities (1.1) to hold on general metric measure spaces with polar decompositions
The main result of this paper is to characterize the weights u and v for which the corresponding Hardy inequality holds on X
Summary
After the Hardy inequality was proved by Hardy in [1], a large amount of literature is available on this inequality. In our previous paper [21], for the case 1 < p ≤ q < ∞, we characterized the weights u and v for the Hardy inequalities (1.1) to hold on general metric measure spaces with polar decompositions. Complementary to [21], we consider the weight characterizations for the case 0 < q < p, 1 < p < ∞ The setting of these papers is rather general, and we consider polarizable metric measure spaces. The main result of this paper is to characterize the weights u and v for which the corresponding Hardy inequality holds on X. Let X be a metric measure space with a polar decomposition at a. Let u, v > 0 be measurable and positive a.e in X such that u ∈ L1(X\{a}) and v1−p ∈ L1loc(X)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
