According to Wikipedia,
However, many results from classical harmonic analysis and computational geometry extend to the setting of metric spaces with doubling measures.
My question is: what are some examples of these results, and where can a more thorough explanation of why this is the case be found?
The question is a bit unclear, so I'm not sure this is what you seek. But here is a connection to computational geoemtry.
There is considerable literature on geometric spanners and doubling metric spaces. Here is a sampling of references from 2006, 2008, 2019, 2022. And from their references you can locate many more.
Damian, Mirela, Saurav Pandit, and Sriram Pemmaraju. "Distributed spanner construction in doubling metric spaces." In International Conference on Principles of Distributed Systems, pp. 157-171. Springer, Berlin, Heidelberg, 2006.
Gottlieb, Lee-Ad, and Liam Roditty. "An optimal dynamic spanner for doubling metric spaces." In European Symposium on Algorithms, pp. 478-489. Springer, Berlin, Heidelberg, 2008.
Borradaile, Glencora, Hung Le, and Christian Wulff-Nilsen. "Greedy spanners are optimal in doubling metrics." In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 2371-2379. Society for Industrial and Applied Mathematics, 2019.
Bartal, Yair, Ora Nova Fandina, and Ofer Neiman. "Covering metric spaces by few trees." Journal of Computer and System Sciences 130 (2022): 26-42.
