So I have a working idea on Gaussian-Poincaré Inequality. Namely through the Ornstein-Ullenbeck Generator and Gaussian Integration by parts.
Recently I have stumbled across Sobolev Spaces and have seen there is a Poincaré Inequality defined there as well over an open set $\Omega$ and w.r.t the Lebesgue Measure. This seems more of a Distribution Theory approach which I admit I have just started learning.
My question is is there any connection between the classical poncare inequality defined for functions on a Sobolov Space and the Gaussian-Poincare Inequality.
Any reference or explanation is deeply appreciated.