Prove2Me

`BookProof.StrichartzWave.opL2_potentialOp_apply_eq_mulL2` (W : V → ℝ) (hW : Function.HasTemperateGrowth W) (hmem : MemLp (fun x => (W x : ℂ)) (⊤ : ℝ≥0∞) (volume : Measure V)) (f :

Proved
BookProof.StrichartzWave.opL2_potentialOp_apply_eq_mulL2

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWaveUnboundedPotential.

BookProof.StrichartzWave.opL2_potentialOp_apply_eq_mulL2 (W : V → ℝ) (hW : Function.HasTemperateGrowth W) (hmem : MemLp (fun x => (W x : ℂ)) (⊤ : ℝ≥0∞) (volume : Measure V)) (f : 𝓢(V, ℂ)) : opL2 (potentialOp W) (schwartzEquiv V f) = mulL2 (hmem.toLp _) (f.toLp 2 (volume : Measure V))

Formalization note: Lean 4 identifier BookProof.StrichartzWave.opL2_potentialOp_apply_eq_mulL2.

Preamble
-- Generated from ChapterWaveUnboundedPotential.lean — theorem BookProof.StrichartzWave.opL2_potentialOp_apply_eq_mulL2
import Mathlib
import Definitions.Def_ChapterWaveUnboundedPotential
import Definitions.Def_ChapterStrichartzWave
import Definitions.Def_ChapterWaveBoundedPotential
open BookProof.StrichartzWave



open MeasureTheory SchwartzMap FourierTransform ComplexInnerProductSpace ENNReal

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

variable {ι : Type*} [Fintype ι]
Formal statement
theorem BookProof.StrichartzWave.opL2_potentialOp_apply_eq_mulL2 (W : V → ℝ) (hW : Function.HasTemperateGrowth W)
    (hmem : MemLp (fun x => (W x : ℂ)) (⊤ : ℝ≥0∞) (volume : Measure V)) (f : 𝓢(V, ℂ)) :
    opL2 (potentialOp W) (schwartzEquiv V f)
      = mulL2 (hmem.toLp _) (f.toLp 2 (volume : Measure V)) := by sorry

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