Bonami–Beckner hypercontractivity on the uniform cube

TheoremVector two-point inequality for nonnegative noise#✓fully proved

The vector two-point inequality holds for 0 ≤ ρ ≤ 1.

theorem vtp_vectorTwoPoint {ρ : ℝ} (h0 : 0 ≤ ρ) (h1 : ρ ≤ 1) :
    bnt_VectorTwoPoint ρ

Uses 5 · Used by 2

Statement usesVector-valued two-point hypercontractive inequality

Proof uses\(L^q\) norm under the uniform measure on the cube · q-th moment under the uniform measure on the cube · Uniform expectation on the Boolean cube · Two-point Bonami inequality

Used byBonami–Beckner hypercontractive inequality · Bonami–Beckner hypercontractive inequality

TheoremBonami–Beckner hypercontractive inequality#✓fully proved

The Bonami–Beckner hypercontractive inequality (Bonami 1970; Beckner 1975): for 0 ≤ ρ ≤ 1 and f : (Fin n → Bool) → ℝ, ‖T_ρ f‖₂ ≤ ‖f‖_(1 + ρ²) under the uniform measure.

theorem vtp_hypercontractivity {ρ : ℝ} (h0 : 0 ≤ ρ) (h1 : ρ ≤ 1)
    (n : ℕ) (f : (Fin n → Bool) → ℝ) :
    bnt_qnorm 2 (bnt_noiseOp ρ f) ≤ bnt_qnorm (1 + ρ ^ 2) f

Uses 4

Statement usesNoise operator on the Boolean cube · \(L^q\) norm under the uniform measure on the cube

Proof usesHypercontractivity from the vector two-point inequality · Vector two-point inequality for nonnegative noise

2 fully proved

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