# Poincare type inequalities via 1-dimensional Malliavin calculus

Shlomo Levental (Michigan State University), *March 21, 2019*

** Abstract: ** We will review briefly 3 types of operators which are mapping spaces of real-valued functions which are defined on the real line equipped with standard normal probability measure. Those are the derivative, divergence and Ornstein-Uklenbeck operators. There are simple formulas that describe the relationships between those operators. Using those formulas the proofs of the following will be presented: 1. Poincare inequality : The variance of a function of N(0,1) is dominated by the second moment of its derivative. 2. An upper bound to the Wasserstein distance between the distribution of a function of N(0,1) (the function has mean 0 and standard deviation 1) and N(0,1) itself. This upper bound is (up to a constant) the multiplication of the L4 norm of the function derivative and the L4 norm of the function 2nd derivative. The material is based on Nourdin and Peccati book.