This talk is devoted to the study of weighted Poincaré inequalities on hyperbolic space, where the weight functions depend on a scaling parameter. This leads to a new family of scale-dependent Poincaré inequalities with Gaussian type measure on the hyperbolic space. As a result, we derive both scale-dependent and scale-invariant $L^2$-stability results for the Heisenberg uncertainty principle.

The video of this talk is available on the IISc Math Department channel.