Scale-Dependent Poincar\'{e} inequalities, log-Sobolev inequality and the stability of the Heisenberg Uncertainty Principle on the hyperbolic space

Abstract

We establish a general scale-dependent Poincar\'{e}-Hardy type identity involving a vector field on the hyperbolic space. By choosing suitable parameter, potential and vector field in this identity, we can recover, as well as derive new versions of and substantially improve several Poincar\'{e} type, Hardy type and Poincar\'{e}-Hardy type inequalities in the literature. We also investigate weighted Poincar\'{e} inequalities on hyperbolic space, where the weight functions depend on a scaling parameter. This leads to a new family of scale-dependent Poincar\'{e} inequalities with Gaussian type measure on the hyperbolic space which is of independent interest. As a result, we derive both scale-dependent and scale-invariant $L^{2}$-stability results for the Heisenberg uncertainty principle in this setting. Finally, we study the logarithmic Sobolev inequality with Gaussian measure on the hyperbolic spaces, that is still missing in the literature.