Third Moment Bound for the Supremum Norm from Coordinates

Proved

LevyStochCalc.Brownian.Ito.integral_norm_pi_pow_three_le_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concavevendored-levystochcalc

This is a finite-dimensional norm estimate.

Let $x:\Omega\to(\operatorname{Fin}(n)\to\mathbb R)$ be measurable. Assume that each coordinate $|x(\cdot,p)|^3$ is integrable and that numbers $c_p\in\mathbb R$ satisfy

\int_\Omega|x(\omega,p)|^3dP(\omega)\le c_p

for every $p\in\operatorname{Fin}(n)$ . Then

\int_\Omega\|x(\omega)\|^3dP(\omega) \le n^2\sum_{p<n}c_p.

This lemma converts coordinatewise cubic estimates into a bound for the norm on $\operatorname{Fin}(n)\to\mathbb R$ and does not require $P$ to be a probability measure.

Original Lean declaration header

Within namespace LevyStochCalc.Brownian.Ito. Shown in its original module context. Proof omitted.

theorem integral_norm_pi_pow_three_le {n : ℕ} {x : Ω → Fin n → ℝ} (hxm : Measurable x)
    (h3 : ∀ p : Fin n, Integrable (fun ω => |x ω p| ^ 3) P) {c : Fin n → ℝ}
    (hc : ∀ p : Fin n, ∫ ω, |x ω p| ^ 3 ∂P ≤ c p) :
    ∫ ω, ‖x ω‖ ^ 3 ∂P ≤ (n : ℝ) ^ 2 * ∑ p, c p

Module navigation

Module: LevyStochCalc/Brownian/VectorItoThirdMoment.

Previous: integral_abs_sum_pow_three_le · Next: integral_abs_vectorItoMartingale_sub_pow_three_le.

Source excerpt, lines 69–72 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoThirdMoment

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Brownian/VectorItoThirdMoment.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
omit [IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.Ito.integral_norm_pi_pow_three_le_readable {n : ℕ} {x : Ω → Fin n → ℝ} (hxm : Measurable x)
    (h3 : ∀ p : Fin n, Integrable (fun ω => |x ω p| ^ 3) P) {c : Fin n → ℝ}
    (hc : ∀ p : Fin n, ∫ ω, |x ω p| ^ 3 ∂P ≤ c p) :
    ∫ ω, ‖x ω‖ ^ 3 ∂P ≤ (n : ℝ) ^ 2 * ∑ p, c p := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoThirdMoment.lean#L67-L88 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoThirdMoment`, declaration `LevyStochCalc.Brownian.Ito.integral_norm_pi_pow_three_le`; the repository is private at upload time). Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (Apache-2.0; upstream copyright notice retained in the Lean text).

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