Poincaré type inequalities on the discrete cube and in the CAR algebra

摘要

We prove L p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {−1, 1} n . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and Götze. We also get inequalities between the L p norms of | Ñf| leftvert nabla frightvert and $Delta ^{alpha }f,alpha > 0;$Delta ^{alpha }f,alpha > 0; moreover L p spaces may be replaced by more general ones. Similar results hold true, replacing functions on the cube by matrices in the *-algebra spanned by n fermions and the L p norm by the Schatten norm C _p .