Cubic Integral Bound for a Sum of Two Functions

Proved

LevyStochCalc.Brownian.Ito.integral_abs_add_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 cubic-moment inequality for two real-valued functions.

Let $u,v:\Omega\to\mathbb R$ be such that $|u|^3$ , $|v|^3$ , and $|u+v|^3$ are integrable with respect to a measure $P$ . Then

\int_\Omega |u(\omega)+v(\omega)|^3\,dP(\omega) \le4\left(\int_\Omega|u(\omega)|^3\,dP(\omega)+\int_\Omega|v(\omega)|^3\,dP(\omega)\right).

This elementary estimate is used to combine drift and martingale third-moment bounds.

Original Lean declaration header

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

theorem integral_abs_add_pow_three_le {P : Measure Ω} {u v : Ω → ℝ}
    (hu : MeasureTheory.Integrable (fun ω => |u ω| ^ 3) P)
    (hv : MeasureTheory.Integrable (fun ω => |v ω| ^ 3) P)
    (huv : MeasureTheory.Integrable (fun ω => |u ω + v ω| ^ 3) P) :
    ∫ ω, |u ω + v ω| ^ 3 ∂P ≤ 4 * ((∫ ω, |u ω| ^ 3 ∂P) + ∫ ω, |v ω| ^ 3 ∂P)

Module navigation

Module: LevyStochCalc/Brownian/ItoFormulaGrid.

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Source excerpt, lines 209–213 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoFormulaGrid

/-
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/ItoFormulaGrid.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]

Formal statement


theorem LevyStochCalc.Brownian.Ito.integral_abs_add_pow_three_le_readable {P : Measure Ω} {u v : Ω → ℝ}
    (hu : MeasureTheory.Integrable (fun ω => |u ω| ^ 3) P)
    (hv : MeasureTheory.Integrable (fun ω => |v ω| ^ 3) P)
    (huv : MeasureTheory.Integrable (fun ω => |u ω + v ω| ^ 3) P) :
    ∫ ω, |u ω + v ω| ^ 3 ∂P ≤ 4 * ((∫ ω, |u ω| ^ 3 ∂P) + ∫ ω, |v ω| ^ 3 ∂P) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoFormulaGrid.lean#L208-L224 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoFormulaGrid`, declaration `LevyStochCalc.Brownian.Ito.integral_abs_add_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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