JournalsrmiVol. 36, No. 1pp. 79–97

Poincaré inequality 3/2 on the Hamming cube

  • Paata Ivanisvili

    University of California, Irvine, USA
  • Alexander Volberg

    Michigan State University, East Lansing, USA
Poincaré inequality 3/2 on the Hamming cube cover

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Abstract

For any n1n \geq 1, and any f ⁣:{1,1}nRf \colon \{-1,1\}^{n} \to \mathbb{R}, we have

RE(f+if)3/2R(Ef)3/2,\mathcal R \mathbb{E} (f + i |\nabla f|)^{3/2} \leq \mathcal R (\mathbb{E}f)^{3/2},

where z3/2z^{3/2} for z=x+iyz=x+iy is taken with principal branch, and R\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