# Poincaré inequality 3/2 on the Hamming cube

### Paata Ivanisvili

University of California, Irvine, USA### Alexander Volberg

Michigan State University, East Lansing, USA

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## Abstract

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.

## Cite this article

Paata Ivanisvili, Alexander Volberg, Poincaré inequality 3/2 on the Hamming cube. Rev. Mat. Iberoam. 36 (2020), no. 1, pp. 79–97

DOI 10.4171/RMI/1122