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

Author： Ben Efraim L.  

Publisher： Springer Publishing Company

ISSN： 0178-8051

Source： Probability Theory and Related Fields, Vol.141, Iss.3-4, 2008-07, pp. : 569-602

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Abstract

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 $$ leftvert abla frightvert $$ and $$Delta ^{alpha }f,alpha > 0;$$ 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 .

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