File size: 5,103 Bytes
ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | 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
end OAI
|