`BookProof.ScalaronHermiteEsa.gaussExpDecay_of_memLp_mul` {u w : Vd d → ℂ} {C c : ℝ} (hu : AEStronglyMeasurable u (volume : Measure (Vd d))) (hw : MemLp w 2 (volume : Measure (Vd d

Proved

BookProof.ScalaronHermiteEsa.gaussExpDecay_of_memLp_mul

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterScalaronHermiteEsa.

BookProof.ScalaronHermiteEsa.gaussExpDecay_of_memLp_mul {u w : Vd d → ℂ} {C c : ℝ} (hu : AEStronglyMeasurable u (volume : Measure (Vd d))) (hw : MemLp w 2 (volume : Measure (Vd d))) (hbd : ∀ x, ‖u x‖ ≤ C * Real.exp (c * ‖x‖) * ‖w x‖) : GaussExpDecay u

Formalization note: Lean 4 identifier BookProof.ScalaronHermiteEsa.gaussExpDecay_of_memLp_mul.

Preamble

-- Generated from ChapterScalaronHermiteEsa.lean — theorem BookProof.ScalaronHermiteEsa.gaussExpDecay_of_memLp_mul
import Definitions.Def_ChapterQgHermiteCore
import Definitions.Def_ChapterQgHermiteFriedrichs
import Definitions.Def_ChapterQgHermiteOscillatorEsa
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterStarobinskyPotential
import Definitions.Def_ChapterStoneBridge
import Definitions.Def_ChapterEsaClosureCore
import Definitions.Def_ChapterSirkTrotterKato
import Mathlib
import Definitions.Def_ChapterScalaronHermiteEsa
import Definitions.Def_ChapterHermiteFunctions
import Definitions.Def_ChapterHermiteProductCore
open BookProof.HermiteCore
open BookProof.HermiteProductCore
open BookProof.ScalaronHermiteEsa



open MeasureTheory Complex MvPolynomial FourierTransform
open BookProof.HermiteProductCore BookProof.QgHermiteCore BookProof.QgHermiteFriedrichs
open BookProof.QgHermiteOscillator BookProof.FarisLavine BookProof.Starobinsky
open BookProof.StoneBridge BookProof.EsaClosure BookProof.ChapterStoneResolvent

noncomputable section

variable {d : ℕ}

Formal statement

theorem BookProof.ScalaronHermiteEsa.gaussExpDecay_of_memLp_mul {u w : Vd d → ℂ} {C c : ℝ}
    (hu : AEStronglyMeasurable u (volume : Measure (Vd d)))
    (hw : MemLp w 2 (volume : Measure (Vd d)))
    (hbd : ∀ x, ‖u x‖ ≤ C * Real.exp (c * ‖x‖) * ‖w x‖) :
    GaussExpDecay u := by sorry

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