I need to use the Lebesgue differentiation theorem for doubling metric measure spaces and was wondering if Carnot groups are doubling. If yes, is there any reference you can point me to? Thank you.



Are Carnot groups (as Carnot Caratheodory metric spaces) doubling?
Carnot groups as well as general (equiregular) SubRiemannian manifolds are typical examples of Ahlfors regular metric measure spaces, which are also geodesic. In particular, they are doubling as a metric space. You can find these facts in the lectures of Heinonen


Heinonen, Juha Calculus on Carnot groups. Fall School in Analysis (Jyväskylä, 1994), 1–31, Report, 68, Univ. Jyväskylä, Jyväskylä, 1995. 


or 


<https://65ac2831-a-62cb3a1a-s-sites.googlegroups.com/site/enricoledonne/LeDonne_subRiemannian.pdf?attachauth=ANoY7crEFlR3ewvIR3hk50AckSkkkGYal_BLGwk8ikRZHg1WCZYVLgvNqfUPxU5OB6N14PeVoRN1OAdl_nAJBa2VcfLXMSQNHxkeFjsEItt-Omoklk1J8ooqftWKJaTC6a9o5S1HrenJlQ-GPbLaRHE27z_jhGl15t-uGwCqILj5DsO4YKZ7yuL2qAiEL1PPkrJ-P-XJ4dOUgWoGIE8v2xKZUOMWQlCeMPA17d5qpLDhaQWkNnV5Weo%3D&attredirects=0>


