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Fourier level inequalities on the discrete cube via complex hypercontractivity and Riesz transforms

  • Fan Chang
  • Zhenyu Li
  • Guanghui Wang
  • Mingrui Wang
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Abstract

Eldan's level-1:3 inequality is a central tool in his second-order correlation inequalities for increasing Boolean functions. His proof is going to Gaussian space and uses an entropy estimate for second Hermite tensors. We give a direct proof on the discrete hypercube. For increasing real-valued functions f,g:{0,1}n→ℝ, we prove W3[f,g]≥ −2W1[f,g]log (√(Var(f)Var(g))/W1[f,g]). The proof combines complex hypercontractivity with the discrete Riesz transforms. We also give a short proof of a level-1:d inequality obtained by Keller and Kindler. As an application, we recover Eldan's correlation inequality with a slight improvement. If f,g:{0,1}n→{0,1} are increasing and g is antipodal, then Cov(f,g)≥ 1/24 ∑iInfi[f]Infi[g]/(√(log (4e√(Var(f)Var(g))/(∑iInfi[f]Infi[g])))), where the square-root logarithm cannot be improved as shown in the OR--majority example.

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Cite

BibTeXRIS
BibTeX
@misc{chang2026fourier,
  title = {{Fourier level inequalities on the discrete cube via complex hypercontractivity and Riesz transforms}},
  author = {Fan Chang and Zhenyu Li and Guanghui Wang and Mingrui Wang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08289},
  eprint = {2611.08289},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08289},
}
RIS
TY  - UNPB
TI  - Fourier level inequalities on the discrete cube via complex hypercontractivity and Riesz transforms
AU  - Fan Chang
AU  - Zhenyu Li
AU  - Guanghui Wang
AU  - Mingrui Wang
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08289
UR  - https://arxiv.org/abs/2611.08289
AB  - Eldan's level-1:3 inequality is a central tool in his second-order correlation inequalities for increasing Boolean functions. His proof is going to Gaussian space and uses an entropy estimate for second Hermite tensors. We give a direct proof on the discrete hypercube. For increasing real-valued functions f,g:{0,1}^n→ℝ, we prove W_3[f,g]≥ −2W_1[f,g]log (√(Var(f)Var(g))/W_1[f,g]). The proof combines complex hypercontractivity with the discrete Riesz transforms. We also give a short proof of a level-1:d inequality obtained by Keller and Kindler. As an application, we recover Eldan's correlation inequality with a slight improvement. If f,g:{0,1}^n→{0,1} are increasing and g is antipodal, then Cov(f,g)≥ 1/24 ∑_iInf_i[f]Inf_i[g]/(√(log (4e√(Var(f)Var(g))/(∑_iInf_i[f]Inf_i[g])))), where the square-root logarithm cannot be improved as shown in the OR--majority example.
ER  -