# Poincaré inequality 3/2 on the Hamming cube

### Paata Ivanisvili

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

Michigan State University, East Lansing, USA

## Abstract

For any $n≥1$, and any $f:{−1,1}_{n}→R$, we have

$RE(f+i∣∇f∣)_{3/2}≤R(Ef)_{3/2},$

where $z_{3/2}$ for $z=x+iy$ is taken with principal branch, and $R$ 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