Prove2Me

`BookProof.NsSpatialMultiplier.fourier_opL2_firstOrderOp` (c : ι → ℝ) (w : ι → V) (f : 𝓢(V, ℂ)) : l2Fourier V (opL2 (firstOrderOp c w) (schwartzEquiv V f)) = (mulSymbolOp (foSymbol

Proved
BookProof.NsSpatialMultiplier.fourier_opL2_firstOrderOp

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNsSpatialMomentumMultiplier.

BookProof.NsSpatialMultiplier.fourier_opL2_firstOrderOp (c : ι → ℝ) (w : ι → V) (f : 𝓢(V, ℂ)) : l2Fourier V (opL2 (firstOrderOp c w) (schwartzEquiv V f)) = (mulSymbolOp (foSymbolFn c w) (𝓕 f)).toLp 2 (volume : Measure V)

Formalization note: Lean 4 identifier BookProof.NsSpatialMultiplier.fourier_opL2_firstOrderOp.

Preamble
-- Generated from ChapterNsSpatialMomentumMultiplier.lean — theorem BookProof.NsSpatialMultiplier.fourier_opL2_firstOrderOp
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_firstOrderOp (c : ι → ℝ) (w : ι → V) (f : 𝓢(V, ℂ)) :
    l2Fourier V (opL2 (firstOrderOp c w) (schwartzEquiv V f))
      = (mulSymbolOp (foSymbolFn c w) (𝓕 f)).toLp 2 (volume : Measure V) := by sorry

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