The Poincaré inequality is an important result in the theory of Sobolev spaces. It allows to estimate the value of the function using values of its derivatives and geometry of the domain. Such estimates are important in modern methods of the calculus of variations. The Poincaré inequality also has broad use in the theory of subelliptic equations. It allows to evaluate eigenvalues of the subelliptic operator, to derive estimates on the fundamental solution, to prove maximum principle.
For the theory of Sobolev spaces on metric spaces, developments of which are very fruitful in recent years, the Poincaré inequality appears to be a fundamental property. There are many works which start with a suggestion that metric measure space admits an analogue of the Poincaré inequality. For instance, in [1] for getting Sobolev-type estimates on a metric space it is sufficient to have just doubling measure condition and Poincaré type p-inequality.
In the talk we consider equiregular sub-Riemannian spaces, generated by C1-smooth horizontal distribution in the sense of the work [2]. In [3] the properties of the Carnot-Caratheodory metric and corresponding Hausdorff measure on such structures are studied. These results are used in [4] to prove an analogue of the Poincaré inequality and to derive Sobolev-type estimates.
1. Hajłasz P., Koskela P., "Sobolev met Poincaré", Memoirs of AMS. 2000. V. 145, N. 688.
2. Karmanova M., Vodopyanov S. "Geometry of Carnot-Carathéodory spaces, Differentiability, Coarea and Area Formulas", Analysis and Mathematical Physics. Basel: Birkhäuser, 2009. P. 284–387.
3. Basalaev S. G., Vodopyanov S. K., "Approximate differentiability of mappings of Carnot-Carathéodory spaces", Eurasian Mathematical Journal. 2013. V. 4, N. 2. P. 10–48.
4. Basalaev S. G., "The Poincaré inequality for C1,α-smooth vector fields", Siberian Mathematical Journal. 2014. V. 55, N. 2. P. 215–229.