`BookProof.ChapterLinftyMaximalAbelian.symbol_mul` {T : Lp ℂ 2 μ →L[ℂ] Lp ℂ 2 μ} (hT : CommutesWithMultOps T) (φ : α → ℂ) (hφ : MemLp φ ⊤ μ) : ((T (multOp φ hφ (oneLp μ)))...

Proved

BookProof.ChapterLinftyMaximalAbelian.symbol_mul

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLinftyMaximalAbelian.

BookProof.ChapterLinftyMaximalAbelian.symbol_mul {T : Lp ℂ 2 μ →L[ℂ] Lp ℂ 2 μ} (hT : CommutesWithMultOps T) (φ : α → ℂ) (hφ : MemLp φ ⊤ μ) : ((T (multOp φ hφ (oneLp μ))) : α → ℂ) =ᵐ[μ] fun x => φ x * symbol T x

Formalization note: Lean 4 identifier BookProof.ChapterLinftyMaximalAbelian.symbol_mul.

Preamble

-- Generated from ChapterLinftyMaximalAbelian.lean — theorem BookProof.ChapterLinftyMaximalAbelian.symbol_mul
import Mathlib
import Definitions.Def_ChapterLinftyMaximalAbelian
import Definitions.Def_ChapterLinftyMultiplication
open BookProof.ChapterLinftyMultiplication
open BookProof.ChapterLinftyMaximalAbelian


noncomputable section

open MeasureTheory ENNReal Complex


open BookProof.ChapterLinftyMultiplication

variable {α : Type*} [MeasurableSpace α] {μ : Measure α} [IsFiniteMeasure μ]

Formal statement

theorem BookProof.ChapterLinftyMaximalAbelian.symbol_mul {T : Lp ℂ 2 μ →L[ℂ] Lp ℂ 2 μ} (hT : CommutesWithMultOps T)
    (φ : α → ℂ) (hφ : MemLp φ ⊤ μ) :
    ((T (multOp φ hφ (oneLp μ))) : α → ℂ) =ᵐ[μ] fun x => φ x * symbol T x := by sorry

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