We prove local refined versions of Hardy's and Rellich's inequalities as well as of uncertainty principles for sums of squares of vector fields on bounded sets of smooth manifolds under certain assumptions on the vector fields. We also give some explicit examples, in particular, for sums of squares of vector fields on Euclidean spaces and for sub-Laplacians on stratified Lie groups.
The authors were supported in parts by the EPSRC grant EP/K039407/1 and by the Leverhulme
Grant RPG-2014-02, as well as by the MESRK grant 5127/GF4. No new data was collected or
generated during the course of the research.
"Local Hardy and Rellich inequalities for sums of squares of vector fields." Adv. Differential Equations 22 (7/8) 505 - 540, July/August 2017.