| import Mathlib |
| import OAI.Analysis.Crouzeix.AngularMoments |
|
|
| namespace OAI |
|
|
| /-! Angular Measure. -/ |
|
|
| noncomputable section |
|
|
| open Set Filter Metric Topology Function Complex ComplexConjugate MeasureTheory |
|
|
| namespace CrouzeixHilbert.Conformal |
|
|
| open Boundary |
|
|
| local instance : Fact (0 < (1 : ℝ)) := ⟨by norm_num⟩ |
|
|
| namespace ExteriorCollar |
|
|
| variable {U : Set ℂ} (C : ExteriorCollar U) (R : InteriorCollar U) |
|
|
| def interiorAngle : C(CircleSpace, CircleSpace) := |
| ⟨fun t => (AddCircle.homeomorphCircle (by norm_num : (1 : ℝ) ≠ 0)).symm |
| ⟨C.interiorBoundary R t, by exact mem_sphere_zero_iff_norm.mpr (C.norm_interiorBoundary R t)⟩, |
| (AddCircle.homeomorphCircle (by norm_num : (1 : ℝ) ≠ 0)).symm.continuous.comp |
| ((C.interiorBoundary R).continuous.subtype_mk _)⟩ |
|
|
| lemma circleCoordinate_interiorAngle (t : CircleSpace) : |
| circleCoordinate (C.interiorAngle R t) = C.interiorBoundary R t := by |
| simp only [circleCoordinate, fourier_one] |
| rw [← AddCircle.homeomorphCircle_apply (by norm_num : (1 : ℝ) ≠ 0)] |
| exact congrArg Subtype.val ((AddCircle.homeomorphCircle (by norm_num : (1 : ℝ) ≠ 0)).apply_symm_apply _) |
|
|
| lemma fourier_interiorAngle (j : ℤ) (t : CircleSpace) : |
| fourier j (C.interiorAngle R t) = (C.interiorBoundary R t)^j := by |
| rw [fourier_apply, AddCircle.toCircle_zsmul, Circle.coe_zpow] |
| have he := C.circleCoordinate_interiorAngle R t |
| simp only [circleCoordinate, fourier_one] at he |
| rw [he] |
|
|
| def angularIntegralCLM : C(CircleSpace, ℂ) →L[ℂ] ℂ := by |
| have hi (f : C(CircleSpace, ℂ)) : Integrable |
| (fun t => C.angularJacobian R t * f (C.interiorAngle R t)) circleMeasure := |
| ((C.angularJacobian R).continuous.mul (f.continuous.comp (C.interiorAngle R).continuous)).integrable_of_hasCompactSupport |
| (HasCompactSupport.of_compactSpace _) |
| let L : C(CircleSpace, ℂ) →ₗ[ℂ] ℂ := { |
| toFun f := ∫ t, C.angularJacobian R t * f (C.interiorAngle R t) ∂circleMeasure |
| map_add' f g := by |
| simp only [ContinuousMap.add_apply, mul_add] |
| exact integral_add (hi f) (hi g) |
| map_smul' c f := by |
| simp only [ContinuousMap.smul_apply, smul_eq_mul, RingHom.id_apply] |
| simp_rw [mul_left_comm _ c, ← smul_eq_mul c] |
| exact integral_smul c _ } |
| refine L.mkContinuous ‖C.angularJacobian R‖ fun f => ?_ |
| have hb : ∀ᵐ t ∂circleMeasure, ‖C.angularJacobian R t * f (C.interiorAngle R t)‖ ≤ |
| ‖C.angularJacobian R‖ * ‖f‖ := Eventually.of_forall fun t => by |
| rw [norm_mul] |
| exact mul_le_mul ((C.angularJacobian R).norm_coe_le_norm t) |
| (f.norm_coe_le_norm _) (norm_nonneg _) (norm_nonneg _) |
| simpa only [L, LinearMap.coe_mk, AddHom.coe_mk, probReal_univ, one_mul, mul_one] using norm_integral_le_of_norm_le_const hb |
|
|
| def circleIntegralCLM : C(CircleSpace, ℂ) →L[ℂ] ℂ := by |
| let L : C(CircleSpace, ℂ) →ₗ[ℂ] ℂ := { |
| toFun f := ∫ t, f t ∂circleMeasure |
| map_add' f g := integral_add |
| (f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) |
| (g.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) |
| map_smul' c f := integral_smul c _ } |
| refine L.mkContinuous 1 fun f => ?_ |
| simpa only [L, LinearMap.coe_mk, AddHom.coe_mk, probReal_univ, one_mul, mul_one] using norm_integral_le_of_norm_le_const |
| (Eventually.of_forall f.norm_coe_le_norm : ∀ᵐ t ∂circleMeasure, ‖f t‖ ≤ ‖f‖) |
|
|
| lemma angularIntegralCLM_eq (hU : IsOpen U) (hc : Convex ℝ U) : |
| C.angularIntegralCLM R = circleIntegralCLM := by |
| apply ContinuousLinearMap.ext_on (s := range (fourier (T := (1 : ℝ)))) |
| · rw [dense_iff_closure_eq, ← Submodule.topologicalClosure_coe, |
| span_fourier_closure_eq_top, Submodule.top_coe] |
| · rintro f ⟨j,rfl⟩ |
| change (∫ t, C.angularJacobian R t * fourier j (C.interiorAngle R t) ∂circleMeasure) = |
| ∫ t, fourier j t ∂circleMeasure |
| simp_rw [C.fourier_interiorAngle] |
| rw [C.integral_angularJacobian_zpow R hU hc, Disk.integral_fourier] |
|
|
| lemma integral_angularChange {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] |
| [CompleteSpace E] (hU : IsOpen U) (hc : Convex ℝ U) (f : C(CircleSpace, E)) : |
| (∫ t, C.angularJacobian R t • f (C.interiorAngle R t) ∂circleMeasure) = |
| ∫ t, f t ∂circleMeasure := by |
| apply (SeparatingDual.eq_iff_forall_dual_eq (R := ℂ)).mpr |
| intro l |
| have h1 : Integrable (fun t => C.angularJacobian R t • f (C.interiorAngle R t)) circleMeasure := |
| ((C.angularJacobian R).continuous.smul (f.continuous.comp (C.interiorAngle R).continuous)).integrable_of_hasCompactSupport |
| (HasCompactSupport.of_compactSpace _) |
| have h2 : Integrable f circleMeasure := f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| rw [← l.integral_comp_comm h1, ← l.integral_comp_comm h2] |
| simp only [map_smul, smul_eq_mul] |
| exact congrArg (fun L : C(CircleSpace, ℂ) →L[ℂ] ℂ => L ⟨l ∘ f, l.continuous.comp f.continuous⟩) |
| (C.angularIntegralCLM_eq R hU hc) |
|
|
| end ExteriorCollar |
|
|
| end CrouzeixHilbert.Conformal |
|
|
| end |
|
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| end OAI |
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|
