Theorems · Theorem · functional analysis

SchwartzMap.integral_bilinear_laplacian_right_eq_left

∀ {E : Type u_5} {F₁ : Type u_9} {F₂ : Type u_10} {F₃ : Type u_11} [inst : NormedAddCommGroup E]
  [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F₁]
  [inst_4 : NormedSpace ℝ F₁] [inst_5 : NormedAddCommGroup F₂] [inst_6 : NormedSpace ℝ F₂]
  [inst_7 : NormedAddCommGroup F₃] [inst_8 : NormedSpace ℝ F₃] [inst_9 : MeasurableSpace E]
  {μ : MeasureTheory.Measure E} [BorelSpace E] [μ.IsAddHaarMeasure] (f : SchwartzMap E F₁) (g : SchwartzMap E F₂)
  (L : F₁ →L[ℝ] F₂ →L[ℝ] F₃),
  ∫ (x : E), (L (f x)) ((Laplacian.laplacian g) x) ∂μ = ∫ (x : E), (L ((Laplacian.laplacian f) x)) (g x) ∂μ

Integration by parts of Schwartz functions for the Laplacian. Version for a general bilinear map.

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Cited by

3 results in Mathlib

Foundations

Depth 287 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFiniteDimensionalNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceMeasureTheory.Measure.IsAddHaarMeasure