| import Mathlib |
| import OAI.Analysis.CoulombIonization.FormDomain.L2 |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter Complex TopologicalSpace |
| open scoped Topology InnerProductSpace ENNReal |
| open MeasureTheory Filter Complex |
| open scoped Topology BigOperators ComplexConjugate FourierTransform SchwartzMap ENNReal |
| open MeasureTheory Filter |
| open scoped Topology ContDiff SchwartzMap FourierTransform ENNReal |
| open MeasureTheory Filter |
| open scoped ContDiff InnerProductSpace Topology |
| open MeasureTheory Filter |
| open scoped ENNReal |
| namespace CoulombSobolev.Test |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
|
|
| def compCL (φ : Test F) (e : E ≃L[ℝ] F) : Test E where |
| fn x := φ.fn (e x) |
| smooth := φ.smooth.comp e.contDiff |
| compact := φ.compact.comp_homeomorph e.toHomeomorph |
|
|
| lemma ext_fn {φ η : Test E} (h : φ.fn = η.fn) : φ = η := by |
| cases φ; cases η; cases h; rfl |
|
|
| lemma compCL_deriv (φ : Test F) (e : E ≃L[ℝ] F) (v : E) : |
| (φ.compCL e).deriv v = (φ.deriv (e v)).compCL e := by |
| apply ext_fn |
| funext x |
| change fderiv ℝ (φ.fn ∘ e) x v = _ |
| rw [fderiv_comp x (φ.smooth.differentiable (by simp) _) e.differentiableAt] |
| simp only [ContinuousLinearEquiv.fderiv, ContinuousLinearMap.comp_apply] |
| rfl |
| end CoulombSobolev.Test |
|
|
| end |
|
|
| end OAI |
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|
