Some weighted functional inequalities and concentration related to Stein kernels in dimension one.

We investigate links between the so-called Stein's density approach in dimension one and some functional and concentration inequalities. We show that measures having a nite first moment and a density with connected support satisfy a weighted Poincar é inequality with the weight being the Stein kernel, that indeed exists and is unique in this case. Furthermore, we prove weighted log-Sobolev and asymmetric Brascamp-Lieb type inequalities related to Stein kernels. We also show that existence of a uniformly bounded Stein kernel is su fficient to ensure a positive Cheeger isoperimetric constant. Then we derive new concentration inequalities. In particular, we prove generalized Mills' type inequalities when a Stein kernel is uniformly bounded and sub-gamma concentration for Lipschitz functions of a variable with a sub-linear Stein kernel. More generally, when some exponential moments are nite, the Laplace transform of the random variable of interest is shown to bounded from above by the Laplace transform of the Stein kernel.