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 \( \left\vert \nabla f\right\vert \) 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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Ben Efraim, L., Lust-Piquard, F. Poincaré type inequalities on the discrete cube and in the CAR algebra. Probab. Theory Relat. Fields 141, 569–602 (2008). https://doi.org/10.1007/s00440-007-0094-x
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DOI: https://doi.org/10.1007/s00440-007-0094-x
Keywords
- Primary: 60E15
- Secondary: 43A70
- 47A20
- 46E39