Revista Matemática Iberoamericana


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Volume 36, Issue 1, 2020, pp. 79–97
DOI: 10.4171/rmi/1122

Published online: 2019-09-17

Poincaré inequality 3/2 on the Hamming cube

Paata Ivanisvili[1] and Alexander Volberg[2]

(1) University of California, Irvine, USA
(2) Michigan State University, East Lansing, USA

For any $n \geq 1$, and any $f \colon \{-1,1\}^{n} \to \mathbb{R}$, we have $$\mathcal R \mathbb{E} (f + i |\nabla f|)^{3/2} \leq \mathcal R (\mathbb{E}f)^{3/2},$$ where $z^{3/2}$ for $z=x+iy$ is taken with principal branch, and $\mathcal R$ denotes the real part. We show an application of this inequality: it sharpens a well-known inequality of Beckner.

Keywords: Hamming cube, two-point inequality, Poincaré inequality, log-Sobolev inequality, Sobolev inequality, Beckner inequality, Gaussian measure

Ivanisvili Paata, Volberg Alexander: Poincaré inequality 3/2 on the Hamming cube. Rev. Mat. Iberoam. 36 (2020), 79-97. doi: 10.4171/rmi/1122