Kernel Of The Closed Weighted Gradient

Proved

FunctionalInequalities.KLS.LogConcave.eq_const_mean_of_closedWeightedOperator_eq_zero_readable

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

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This theorem identifies the kernel of the closed unit-weight gradient with constants.

Let $V$ be finite-dimensional, let $m$ be an additive Haar measure, let $\nu$ be log-concave with nonempty interior support, and let $U:V\to\mathbb R$ be continuous with $\nu=m.\operatorname{tilted}(-U)$ . If

h\in D\!\left(\overline{T}_{\nu,m,I}\right) \quad\text{and}\quad \overline{T}_{\nu,m,I}h=0,

then, as an element of $L^2(\nu)$ ,

h=\operatorname{const}_{\nu}\!\left(\int_V h\,d\nu\right).

Here closedWeightedOperator is the closed unit-weight gradient and L2.mean is the $L^2(\nu)$ mean. The conclusion is an operator-kernel characterization, useful in range-density and orthogonality arguments.

Original Lean declaration header

Within namespace FunctionalInequalities.KLS.LogConcave. Shown in its original module context. Proof omitted.

theorem eq_const_mean_of_closedWeightedOperator_eq_zero (hν : LogConcave ν)
    (hi : (interior (ν : Measure V).support).Nonempty) (hνU : (ν : Measure V) = m.tilted (-U))
    (hU : Continuous U)
    (h : (hν.closedWeightedOperator m (ContinuousLinearMap.id ℝ V) hi).domain)
    (h0 : hν.closedWeightedOperator m (ContinuousLinearMap.id ℝ V) hi h = 0) :
    (h : Lp ℝ 2 (ν : Measure V)) = L2.const ν (L2.mean ν (h : Lp ℝ 2 (ν : Measure V)))

Module navigation

Module: KLS/LogConcave/WeightedGradient/Kernel.

Definition modules: L2/MeanVariance, KLS/LogConcave/WeightedGradient/Closure, Probability/LogConcave.

Supporting theorems: ae_add_eq_of_integral_mul_fderiv_eq_zero, exists_ae_eq_const_of_measurable_of_forall_ae_add_eq, integral_mul_fderiv_eq_zero_of_closedWeightedOperator_eq_zero, restrict_le_smul_of_eq_tilted.

Previous: integral_mul_fderiv_eq_zero_of_closedWeightedOperator_eq_zero.

Source excerpt, lines 446–451 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_FunctionalInequalities_KLS_LogConcave_WeightedGradient_Kernel

open MeasureTheory Filter Topology InnerProductSpace
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
  [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] [FiniteDimensional ℝ V]
  {ν : MeasureTheory.ProbabilityMeasure V} (m : MeasureTheory.Measure V) [m.IsAddHaarMeasure] {U : V → ℝ}

Formal statement


theorem FunctionalInequalities.KLS.LogConcave.eq_const_mean_of_closedWeightedOperator_eq_zero_readable (hν : LogConcave ν)
    (hi : (interior (ν : Measure V).support).Nonempty) (hνU : (ν : Measure V) = m.tilted (-U))
    (hU : Continuous U)
    (h : (hν.closedWeightedOperator m (ContinuousLinearMap.id ℝ V) hi).domain)
    (h0 : hν.closedWeightedOperator m (ContinuousLinearMap.id ℝ V) hi h = 0) :
    (h : Lp ℝ 2 (ν : Measure V)) = L2.const ν (L2.mean ν (h : Lp ℝ 2 (ν : Measure V))) := by sorry

Source

Lean source: https://github.com/human-ai-mathematics/kls-lean/blob/408915f/formalization/FunctionalInequalities/KLS/LogConcave/WeightedGradient/Kernel.lean#L444-L470 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.KLS.LogConcave.WeightedGradient.Kernel`, declaration `FunctionalInequalities.KLS.LogConcave.eq_const_mean_of_closedWeightedOperator_eq_zero`; the repository is private at upload time).

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