`BookProof.NsSpatialMultiplier.fourier_opL2_momentumOp` (m : V) (f : 𝓢(V, ℂ)) : l2Fourier V (opL2 (momentumOp m) (schwartzEquiv V f)) = (mulSymbolOp (fun x => 2 * Real.pi * (inner

Proved

BookProof.NsSpatialMultiplier.fourier_opL2_momentumOp

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNsSpatialMomentumMultiplier.

BookProof.NsSpatialMultiplier.fourier_opL2_momentumOp (m : V) (f : 𝓢(V, ℂ)) : l2Fourier V (opL2 (momentumOp m) (schwartzEquiv V f)) = (mulSymbolOp (fun x => 2 * Real.pi * (inner ℝ x m : ℝ)) (𝓕 f)).toLp 2 (volume : Measure V)

Formalization note: Lean 4 identifier BookProof.NsSpatialMultiplier.fourier_opL2_momentumOp.

Preamble

-- Generated from ChapterNsSpatialMomentumMultiplier.lean — theorem BookProof.NsSpatialMultiplier.fourier_opL2_momentumOp
import Mathlib
import Definitions.Def_ChapterNsSpatialMomentumMultiplier
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterFourierMultiplierEsa
import Definitions.Def_ChapterStrichartzWave
open BookProof.FarisLavine
open BookProof.FourierMultiplierEsa
open BookProof.StrichartzWave
open BookProof.NsSpatialMultiplier



open MeasureTheory SchwartzMap FourierTransform
open BookProof.StrichartzWave BookProof.FourierMultiplierEsa

noncomputable section

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

variable (V) in

Formal statement

theorem BookProof.NsSpatialMultiplier.fourier_opL2_momentumOp (m : V) (f : 𝓢(V, ℂ)) :
    l2Fourier V (opL2 (momentumOp m) (schwartzEquiv V f))
      = (mulSymbolOp (fun x => 2 * Real.pi * (inner ℝ x m : ℝ)) (𝓕 f)).toLp
          2 (volume : Measure V) := by sorry

View graph