`BookProof.WeylCauchyRiemann.dbar_mollify` {F : ℂ → ℂ} (hF : LocallyIntegrable F) {χ : ℂ → ℝ} (hχ : ContDiff ℝ ∞ χ) (hχc : HasCompactSupport χ) (z : ℂ) : dbar (mollify F...

Proved

BookProof.WeylCauchyRiemann.dbar_mollify

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWeylCauchyRiemann.

BookProof.WeylCauchyRiemann.dbar_mollify {F : ℂ → ℂ} (hF : LocallyIntegrable F) {χ : ℂ → ℝ} (hχ : ContDiff ℝ ∞ χ) (hχc : HasCompactSupport χ) (z : ℂ) : dbar (mollify F χ) z = ∫ w : ℂ, dbarR χ (z - w) * F w

Formalization note: Lean 4 identifier BookProof.WeylCauchyRiemann.dbar_mollify.

Preamble

-- Generated from ChapterWeylCauchyRiemann.lean — theorem BookProof.WeylCauchyRiemann.dbar_mollify
import Definitions.Def_ChapterRadialMollifier
import Mathlib
import Definitions.Def_ChapterWeylCauchyRiemann
open BookProof.WeylCauchyRiemann



open MeasureTheory Metric Set Filter Complex BookProof.RadialMollifier
open scoped Convolution Topology ContDiff

noncomputable section

Formal statement

theorem BookProof.WeylCauchyRiemann.dbar_mollify {F : ℂ → ℂ} (hF : LocallyIntegrable F) {χ : ℂ → ℝ}
    (hχ : ContDiff ℝ ∞ χ) (hχc : HasCompactSupport χ) (z : ℂ) :
    dbar (mollify F χ) z = ∫ w : ℂ, dbarR χ (z - w) * F w := by sorry

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