L² Product Bound in eLpNorm

Proved

LevyStochCalc.Brownian.Ito.eLpNorm_one_mul_le_readable

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

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

Let $(\Omega,\mu)$ be a measure space and let $f,g:\Omega\to\mathbb R$ be almost-everywhere measurable. Then the extended $L^p$ norms satisfy

\|fg\|_{L^1(\mu)}\le \|f\|_{L^2(\mu)}\,\|g\|_{L^2(\mu)}.

Here each eLpNorm is an extended nonnegative value in $[0,\infty]$ ; no finiteness assumption is imposed. This is the Cauchy–Schwarz/Hölder estimate used to control products in $L^1$ from $L^2$ bounds.

Original Lean declaration header

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

lemma eLpNorm_one_mul_le {μ : MeasureTheory.Measure Ω} {f g : Ω → ℝ}
    (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    MeasureTheory.eLpNorm (f * g) 1 μ
      ≤ MeasureTheory.eLpNorm f 2 μ * MeasureTheory.eLpNorm g 2 μ

Module navigation

Module: LevyStochCalc/Brownian/ItoL2CompletionConvergence.

Next: tendsto_eLpNorm_one_sq_sub.

Source excerpt, lines 112–115 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoL2CompletionConvergence

/-
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/ItoL2CompletionConvergence.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
universe u
variable {Ω : Type u} [MeasurableSpace Ω]

Formal statement


lemma LevyStochCalc.Brownian.Ito.eLpNorm_one_mul_le_readable {μ : MeasureTheory.Measure Ω} {f g : Ω → ℝ}
    (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    MeasureTheory.eLpNorm (f * g) 1 μ
      ≤ MeasureTheory.eLpNorm f 2 μ * MeasureTheory.eLpNorm g 2 μ := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoL2CompletionConvergence.lean#L109-L128 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoL2CompletionConvergence`, declaration `LevyStochCalc.Brownian.Ito.eLpNorm_one_mul_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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