Sharp Gaussian divergence and Bernstein-Markov inequalities

Abstract

We prove two dimension-free estimates in Gaussian space. The first is an optimal Meyer-type inequality for the Gaussian divergence: for p≥2, |δV|_{L^p(γ_d)} ≤ √p |𝔼 V| +Cp |DV|_{L^p(γ_d;HS_d)} . Here HS_d is the space of d× d matrices with its Hilbert-Schmidt norm. Its main ingredient is a dimension-free weak-type (1,1) bound for the second-order transform D²𝒩⁻¹, where 𝒩 := −Δ+x· D is the Ornstein-Uhlenbeck operator. We also include a direct change-of-variables proof of the weaker bounded derivative divergence inequality in the appendix. The second result is a Gaussian Bernstein-Markov inequality. If P is a polynomial, p≥2, and m_p := (𝔼|g|^p)^1/p for a standard Gaussian g, then |DP|_{L^p(γ_d;ℝ^d)} ≤(2√(2e deg(P)))/m_p |P|_{L^p(γ_d)} . The factor $2$ is unnecessary when p is even, or P is even or odd. Thus, the constant is of order √(deg(P)/p) throughout the range p≥2.

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BibTeX RIS

BibTeX

@misc{chen2026sharp,
  title = {{Sharp Gaussian divergence and Bernstein-Markov inequalities}},
  author = {Fan Chen and Sinho Chewi and Jianfeng Lu and Matthew S. Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32851},
  eprint = {2609.32851},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32851},
}

RIS

TY  - UNPB
TI  - Sharp Gaussian divergence and Bernstein-Markov inequalities
AU  - Fan Chen
AU  - Sinho Chewi
AU  - Jianfeng Lu
AU  - Matthew S. Zhang
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32851
UR  - https://arxiv.org/abs/2609.32851
AB  - We prove two dimension-free estimates in Gaussian space. The first is an optimal Meyer-type inequality for the Gaussian divergence: for p≥2, |δV|_L^p(γ_d) ≤ √p |𝔼 V| +Cp |DV|_L^p(γ_d;HS_d) . Here HS_d is the space of d× d matrices with its Hilbert-Schmidt norm. Its main ingredient is a dimension-free weak-type (1,1) bound for the second-order transform D^2𝒩^−1, where 𝒩 := −Δ+x· D is the Ornstein-Uhlenbeck operator. We also include a direct change-of-variables proof of the weaker bounded derivative divergence inequality in the appendix. The second result is a Gaussian Bernstein-Markov inequality. If P is a polynomial, p≥2, and m_p := (𝔼|g|^p)^1/p for a standard Gaussian g, then |DP|_L^p(γ_d;ℝ^d) ≤(2√(2e deg(P)))/m_p |P|_L^p(γ_d) . The factor $2$ is unnecessary when p is even, or P is even or odd. Thus, the constant is of order √(deg(P)/p) throughout the range p≥2.
ER  -

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