Affine Oscillation Bound for Coordinate Second Derivatives

Proved

LevyStochCalc.abs_coordDeriv_two_sub_le_affine_readable

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

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

Let n : ℕ, let f″ assign a continuous bilinear map on Fin n → ℝ to each point, and let A, K : ℝ. Assume that for all z, w : Fin n → ℝ, ‖f″(z) − f″(w)‖ ≤ A + K‖z − w‖. Then for every coordinate pair p, q : Fin n and points x, y : Fin n → ℝ, the coordinate second derivative coordDeriv₂ f″ p q satisfies

|\operatorname{coordDeriv}_2(f'',p,q,x)-\operatorname{coordDeriv}_2(f'',p,q,y)|\le A+K\|x-y\|.

This technical lemma transfers an operator-norm oscillation estimate to individual coordinate entries.

Original Lean declaration header

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

theorem abs_coordDeriv₂_sub_le_affine {A K : ℝ} (h : ∀ z w, ‖f'' z - f'' w‖ ≤ A + K * ‖z - w‖)
    (p q : Fin n) (x y : Fin n → ℝ) :
    |coordDeriv₂ f'' p q x - coordDeriv₂ f'' p q y| ≤ A + K * ‖x - y‖

Module navigation

Module: LevyStochCalc/Brownian/CoordDerivative.

Previous: norm_le_sum_abs_apply₂.

Source excerpt, lines 205–207 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_CoordDerivative

/-
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/CoordDerivative.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
variable {n : ℕ}
variable {f' : (Fin n → ℝ) → (Fin n → ℝ) →L[ℝ] ℝ}
  {f'' : (Fin n → ℝ) → (Fin n → ℝ) →L[ℝ] (Fin n → ℝ) →L[ℝ] ℝ}

Formal statement


theorem LevyStochCalc.abs_coordDeriv_two_sub_le_affine_readable {A K : ℝ} (h : ∀ z w, ‖f'' z - f'' w‖ ≤ A + K * ‖z - w‖)
    (p q : Fin n) (x y : Fin n → ℝ) :
    |coordDeriv₂ f'' p q x - coordDeriv₂ f'' p q y| ≤ A + K * ‖x - y‖ := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/CoordDerivative.lean#L204-L218 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.CoordDerivative`, declaration `LevyStochCalc.abs_coordDeriv₂_sub_le_affine`; 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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