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Registry leaf card · analysis.calculus.pilp-radial-cutoff-gradient-L1-tendsto-zero

tendsto_integral_norm_fderiv_radialSmoothCutoff_comp_toLp_apply

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- For an integrable finite Pi-space vector field, the `L¹` norm of the radial-cutoff gradient applied to that field vanishes as the cutoff scale tends to infinity. The domination retains the operator norm of the inverse `PiLp` equivalence: the raw Pi norm is not identified with the Euclidean `L²` norm. This theorem only controls the cutoff-gradient cross term; it proves no source-field integrability, main-term convergence, integration by parts, or invariant-law statement.

Plain-English statement

- For an integrable finite Pi-space vector field, the `L¹` norm of the radial-cutoff gradient applied to that field vanishes as the cutoff scale tends to infinity. The domination retains the operator norm of the inverse `PiLp` equivalence: the raw Pi norm is not identified with the Euclidean `L²` norm. This theorem only controls the cutoff-gradient cross term; it proves no source-field integrability, main-term convergence, integration by parts, or invariant-law statement.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.
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Lean statement

theorem tendsto_integral_norm_fderiv_radialSmoothCutoff_comp_toLp_apply
    {n : ℕ} {μ : Measure (Fin (n + 1) → ℝ)}
    {G : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ}
    (hG : Integrable G μ) :
    Tendsto
      (fun R : ℝ =>
        ∫ x, ‖fderiv ℝ
          (fun z => Cutoff.radialSmoothCutoff R
            (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1))))
          x (G x)‖ ∂μ)
      atTop (𝓝 0) := by
  let e : EuclideanSpace ℝ (Fin (n + 1)) ≃L[ℝ] (Fin (n + 1) → ℝ) :=
    PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin (n + 1) => ℝ)
  obtain ⟨C, hC_pos, hC⟩ :=
    Cutoff.radialSmoothCutoff_fderiv_bound
      (E := EuclideanSpace ℝ (Fin (n + 1)))
  let bound : (Fin (n + 1) → ℝ) → ℝ :=
    fun x => (C * ‖e.symm.toContinuousLinearMap‖) * ‖G x‖
  have hbound_integrable : Integrable bound μ := by
    exact (hG.norm.const_mul (C * ‖e.symm.toContinuousLinearMap‖))
  have hmeas :
      ∀ᶠ R : ℝ in atTop,
        AEStronglyMeasurable
          (fun x => ‖fderiv ℝ
            (fun z => Cutoff.radialSmoothCutoff R
              (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1))))
            x (G x)‖) μ := by
    filter_upwards [eventually_gt_atTop (0 : ℝ)] with R hR
    have hsmooth :
        ContDiff ℝ (⊤ : ℕ∞)
          (fun z : Fin (n + 1) → ℝ =>
            Cutoff.radialSmoothCutoff R
              (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1)))) :=
      (Cutoff.radialSmoothCutoff_contDiff hR).comp
        (PiLp.contDiff_toLp (𝕜 := ℝ) (E := fun _ : Fin (n + 1) => ℝ))
    have hderiv :
        AEStronglyMeasurable
          (fun x => fderiv ℝ
            (fun z : Fin (n + 1) → ℝ =>
              Cutoff.radialSmoothCutoff R
                (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1))))
            x) μ :=
      (hsmooth.continuous_fderiv
        (WithTop.coe_ne_zero.mpr WithTop.top_ne_zero)).aestronglyMeasurable
    let eval :
        ((Fin (n + 1) → ℝ) →L[ℝ] ℝ) →L[ℝ]
          (Fin (n + 1) → ℝ) →L[ℝ] ℝ :=
      ContinuousLinearMap.flip (ContinuousLinearMap.apply ℝ ℝ)
    exact
      (eval.aestronglyMeasurable_comp₂
        hderiv hG.aestronglyMeasurable).norm
  have hdom :
      ∀ᶠ R : ℝ in atTop, ∀ᵐ x ∂μ,
        ‖(fun x => ‖fderiv ℝ
          (fun z => Cutoff.radialSmoothCutoff R
            (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1))))
          x (G x)‖) x‖ ≤ bound x := by
    filter_upwards [eventually_ge_atTop (1 : ℝ)] with R hR
    filter_upwards with x
    have hR_pos : 0 < R := lt_of_lt_of_le zero_lt_one hR
    have hfderiv :
        fderiv ℝ
          (fun z => Cutoff.radialSmoothCutoff R
            (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1))))
          x =
        (fderiv ℝ
          (Cutoff.radialSmoothCutoff R :
            EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
          (WithLp.toLp 2 x)).comp e.symm.toContinuousLinearMap :=
      (hasFDerivAt_radialSmoothCutoff_comp_toLp hR_pos x).fderiv
    rw [norm_norm, hfderiv]
    calc
      ‖((fderiv ℝ
          (Cutoff.radialSmoothCutoff R :
            EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
          (WithLp.toLp 2 x)).comp e.symm.toContinuousLinearMap) (G x)‖
          ≤ ‖(fderiv ℝ
              (Cutoff.radialSmoothCutoff R :
                EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
              (WithLp.toLp 2 x)).comp e.symm.toContinuousLinearMap‖ * ‖G x‖ :=
        ContinuousLinearMap.le_opNorm _ _
      _ ≤ ((C / R) * ‖e.symm.toContinuousLinearMap‖) * ‖G x‖ := by
        gcongr
        exact (ContinuousLinearMap.opNorm_comp_le _ _).trans
          (mul_le_mul_of_nonneg_right
            (hC R hR_pos (WithLp.toLp 2 x)) (norm_nonneg _))
      _ ≤ bound x := by
        dsimp [bound]
        gcongr
        exact div_le_self hC_pos.le hR
  have hpoint :
      ∀ᵐ x ∂μ,
        Tendsto
          (fun R : ℝ => ‖fderiv ℝ
            (fun z => Cutoff.radialSmoothCutoff R
              (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1))))
            x (G x)‖)
          atTop (𝓝 0) := by
    filter_upwards with x
    refine squeeze_zero'
      (g := fun R =>
        ((C / R) * ‖e.symm.toContinuousLinearMap‖) * ‖G x‖) ?_ ?_ ?_
    · exact Filter.Eventually.of_forall fun R => norm_nonneg _
    · filter_upwards [eventually_gt_atTop (0 : ℝ)] with R hR
      have hfderiv :
          fderiv ℝ
            (fun z => Cutoff.radialSmoothCutoff R
              (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1))))
            x =
          (fderiv ℝ
            (Cutoff.radialSmoothCutoff R :
              EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
            (WithLp.toLp 2 x)).comp e.symm.toContinuousLinearMap :=
        (hasFDerivAt_radialSmoothCutoff_comp_toLp hR x).fderiv
      rw [hfderiv]
      calc
        ‖((fderiv ℝ
            (Cutoff.radialSmoothCutoff R :
              EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
            (WithLp.toLp 2 x)).comp e.symm.toContinuousLinearMap) (G x)‖
            ≤ ‖(fderiv ℝ
                  (Cutoff.radialSmoothCutoff R :
                    EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
                  (WithLp.toLp 2 x)).comp e.symm.toContinuousLinearMap‖ *
                  ‖G x‖ := ContinuousLinearMap.le_opNorm _ _
        _ ≤ ((C / R) * ‖e.symm.toContinuousLinearMap‖) * ‖G x‖ := by
          gcongr
          exact (ContinuousLinearMap.opNorm_comp_le _ _).trans
            (mul_le_mul_of_nonneg_right
              (hC R hR (WithLp.toLp 2 x)) (norm_nonneg _))
    · simpa [mul_assoc] using
        (tendsto_const_nhds.div_atTop tendsto_id).mul_const
          (‖e.symm.toContinuousLinearMap‖ * ‖G x‖)
  have hDCT :=
    MeasureTheory.tendsto_integral_filter_of_dominated_convergence
      (μ := μ) (l := atTop)
      (F := fun R x => ‖fderiv ℝ
        (fun z => Cutoff.radialSmoothCutoff R
          (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1))))
        x (G x)‖)
      (f := fun _ => (0 : ℝ)) bound hmeas hdom hbound_integrable hpoint
  simpa using hDCT

/-- Multiplication by the PiLp-wrapped radial cutoff converges to the identity
under integration for every integrable real normed-space-valued source field.

The statement is measure-generic and uses only integrability of the source.
It proves the cutoff main-term limit, but no Gibbs-specific integrability,
cutoff-gradient estimate, integration by parts, generator-domain result, or
invariant-law statement. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:157published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 cutoff-smul route: make the cutoff-gradient cross term vanish in L1 for every integrable finite-Pi vector field

Lean proof walkthrough

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Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

Hidden assumptions and non-claims

  • Measurability is represented explicitly or must be supplied by a dependency.
  • Integrability is an input or proved output; a displayed integral alone does not supply it.
  • Totalized `fderiv` values must not be read as a differentiability theorem.
  • Almost-everywhere hypotheses depend on the stated measure and representative.
  • A Gibbs expression is not automatically a probability law or an invariant law.
  • A formal generator display does not establish a closed operator domain.
  • A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.
Common pitfall. A wrapper or display identity is not a new analytic theorem merely because it has its own Lean name. Check the statement, hypotheses, and downstream consumers before interpreting its mathematical contribution.