`BookProof.MixedLinearEsa.momentumOp_cutSchwartz` (m : V) (g : V → ℝ) (hg : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) g) (hgcs : HasCompactSupport g) (f : 𝓢(V, ℂ)) (x : V) :...

Proved

BookProof.MixedLinearEsa.momentumOp_cutSchwartz

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMixedLinearEsa.

BookProof.MixedLinearEsa.momentumOp_cutSchwartz (m : V) (g : V → ℝ) (hg : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) g) (hgcs : HasCompactSupport g) (f : 𝓢(V, ℂ)) (x : V) : (momentumOp m (cutSchwartz g hg hgcs f)) x = -Complex.I * (f x * ((fderiv ℝ g x m : ℝ) : ℂ)) + ((g x : ℝ) : ℂ) * (momentumOp m f x)

Formalization note: Lean 4 identifier BookProof.MixedLinearEsa.momentumOp_cutSchwartz.

Preamble

-- Generated from ChapterMixedLinearEsa.lean — theorem BookProof.MixedLinearEsa.momentumOp_cutSchwartz
import Definitions.Def_ChapterStrichartzWave
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterMixedLinearEsa
import Definitions.Def_ChapterFourierMultiplierEsa
open BookProof.FourierMultiplierEsa
open BookProof.MixedLinearEsa



open MeasureTheory SchwartzMap FourierTransform ComplexInnerProductSpace LineDeriv
open BookProof.StrichartzWave BookProof.FourierMultiplierEsa BookProof.FarisLavine

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V]
  [MeasurableSpace V] [BorelSpace V]

Formal statement

theorem BookProof.MixedLinearEsa.momentumOp_cutSchwartz (m : V) (g : V → ℝ) (hg : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) g)
    (hgcs : HasCompactSupport g) (f : 𝓢(V, ℂ)) (x : V) :
    (momentumOp m (cutSchwartz g hg hgcs f)) x
      = -Complex.I * (f x * ((fderiv ℝ g x m : ℝ) : ℂ))
        + ((g x : ℝ) : ℂ) * (momentumOp m f x) := by sorry

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