lean/OAI/Analysis/IntegralMeans/TransferOperator.lean

import OAI.Analysis.IntegralMeans.TransportBoxes

namespace OAI

noncomputable section
open Set MeasureTheory Filter Function InnerProductSpace
open scoped Topology ComplexConjugate Manifold NNReal ENNReal InnerProductSpace Classical
open MeasureTheory Function
open Set Filter
open Set MeasureTheory Filter Function
open Set MeasureTheory Filter Function InnerProductSpace
open TopologicalSpace
open scoped CompactlySupported
open scoped ENNReal
open scoped Manifold
open scoped Topology CompactlySupported ComplexConjugate
open scoped Topology ComplexConjugate Manifold NNReal ENNReal InnerProductSpace Classical
open scoped Topology ENNReal NNReal
namespace Brennan

attribute [local irreducible] classWeight
attribute [local irreducible] classFun

def halfMul (z w : halfPlane) : halfPlane := ⟨affine z w,affine_mem z.2 w.2⟩

def halfPoint (x y : ℝ) (hy : 0 < y) : halfPlane :=
  ⟨(x : ℂ)+(y : ℂ)*Complex.I,by simpa [halfPlane]⟩

def horizontal (x : ℝ) : halfPlane := halfPoint x 1 zero_lt_one

lemma halfPoint_mul (x y u v : ℝ) (hy : 0 < y) (hv : 0 < v) :
    halfMul (halfPoint x y hy) (halfPoint u v hv) = halfPoint (x+y*u) (y*v) (mul_pos hy hv) := by
  apply Subtype.ext
  apply Complex.ext <;> simp [halfMul,halfPoint,affine]

lemma halfMul_assoc (z w u : halfPlane) : halfMul (halfMul z w) u = halfMul z (halfMul w u) :=
  Subtype.ext (affine_assoc z w u)

lemma halfOne_mul (z : halfPlane) : halfMul halfOne z = z := Subtype.ext (I_affine z)

lemma halfMul_one (z : halfPlane) : halfMul z halfOne = z := Subtype.ext (affine_I z)

lemma continuous_halfPoint.{u_1} {Y : Type u_1} [TopologicalSpace Y] {x y : Y → ℝ}
    (hx : Continuous x) (hy : Continuous y) (hp : ∀ a, 0 < y a) :
    Continuous (fun a => halfPoint (x a) (y a) (hp a)) :=
  ((Complex.continuous_ofReal.comp hx).add ((Complex.continuous_ofReal.comp hy).mul continuous_const)).subtype_mk _

lemma rerootClass_one (g : DiskClass) : rerootClass g halfOne = g := by
  apply class_ext
  intro z hz
  rw [classFun_rerootClass _ _ hz]
  simp [reroot,halfOne,(classFun_schlicht g).2.1,(classFun_schlicht g).2.2]

lemma classWeight_one (g : DiskClass) : classWeight g halfOne = 1 := by
  simp [classWeight,reciprocalDeriv,halfOne,(classFun_schlicht g).2.2]

attribute [local irreducible] rerootClass

def weightAt (z : halfPlane) : C(DiskClass,ℝ) :=
  ⟨fun g => classWeight g z,by
    have hc : Continuous (fun g : DiskClass => (g,z)) := continuous_id.prodMk continuous_const
    exact continuous_classWeight.comp hc⟩

def actionAt (z : halfPlane) : C(DiskClass,DiskClass) :=
  ⟨fun g => rerootClass g z,by
    have hc : Continuous (fun g : DiskClass => (g,z)) := continuous_id.prodMk continuous_const
    exact continuous_rerootClass.comp hc⟩

def affineOperator (z : halfPlane) : C(DiskClass,ℝ) →L[ℝ] C(DiskClass,ℝ) :=
  (ContinuousLinearMap.mul ℝ C(DiskClass,ℝ) (weightAt z)).comp (ContinuousMap.compCLM ℝ ℝ (actionAt z))

lemma affineOperator_apply (z : halfPlane) (f : C(DiskClass,ℝ)) (g : DiskClass) :
    affineOperator z f g = classWeight g z * f (rerootClass g z) := rfl

lemma affineOperator_pos (z : halfPlane) (f : C(DiskClass,ℝ)) (hf : 0 ≤ f) :
    0 ≤ affineOperator z f := fun g => mul_nonneg (classWeight_pos g z).le (hf _)

lemma affineOperator_mul (z w : halfPlane) :
    affineOperator z * affineOperator w = affineOperator (halfMul z w) := by
  ext f g
  change classWeight g z * (classWeight (rerootClass g z) w * f (rerootClass (rerootClass g z) w)) = _
  rw [← mul_assoc,classWeight_cocycle,rerootClass_comp]
  rfl

lemma affineOperator_one : affineOperator halfOne = 1 := by
  ext f g
  simp only [affineOperator_apply,classWeight_one,rerootClass_one,one_mul]
  rfl

lemma continuous_affineOperator_apply (f : C(DiskClass,ℝ)) :
    Continuous (fun z : halfPlane => affineOperator z f) := by
  apply ContinuousMap.continuous_of_continuous_uncurry
  have hw : Continuous (fun p : halfPlane × DiskClass => classWeight p.2 p.1) :=
    continuous_classWeight.comp (continuous_swap : Continuous (Prod.swap : halfPlane × DiskClass → DiskClass × halfPlane))
  have ht : Continuous (fun p : halfPlane × DiskClass => rerootClass p.2 p.1) :=
    continuous_rerootClass.comp (continuous_swap : Continuous (Prod.swap : halfPlane × DiskClass → DiskClass × halfPlane))
  exact hw.mul (f.continuous.comp ht)

lemma affineOperator_norm_le {z : halfPlane} {B : ℝ} (hB : 0 ≤ B)
    (hw : ∀ g : DiskClass, classWeight g z ≤ B) : ‖affineOperator z‖ ≤ B := by
  apply (affineOperator z).opNorm_le_bound hB
  intro f
  apply (ContinuousMap.norm_le _ (mul_nonneg hB (norm_nonneg f))).mpr
  intro g
  rw [affineOperator_apply,norm_mul,Real.norm_of_nonneg (classWeight_pos g z).le]
  exact mul_le_mul (hw g) (ContinuousMap.norm_coe_le_norm f _) (norm_nonneg _) hB

lemma classFun_identityClass : EqOn (classFun identityClass) identityHalf halfPlane :=
  classFun_classOfHalf identityHalf_schlicht

lemma classWeight_identityClass (z : halfPlane) : classWeight identityClass z = 1 := by
  have hd := classFun_identityClass.deriv isOpen_halfPlane z.2
  have hi : deriv identityHalf z = 1 := ((hasDerivAt_id (z : ℂ)).sub_const Complex.I).deriv
  simp [classWeight,reciprocalDeriv,hd,hi]

lemma reroot_identityClass (z : halfPlane) : rerootClass identityClass z = identityClass := by
  apply class_ext
  intro w hw
  rw [classFun_rerootClass _ _ hw,reroot_congr classFun_identityClass z.2 hw,classFun_identityClass hw]
  have hd : deriv identityHalf z = 1 := ((hasDerivAt_id (z : ℂ)).sub_const Complex.I).deriv
  have he : (z.val.re : ℂ)+(z.val.im : ℂ)*Complex.I = z := by apply Complex.ext <;> simp
  dsimp [reroot,identityHalf,affine]
  rw [hd,mul_one]
  apply (div_eq_iff (Complex.ofReal_ne_zero.mpr (ne_of_gt z.2))).mpr
  linear_combination he

def dyadicLeft : halfPlane := halfPoint (-1/2) (1/2) (by norm_num)

def dyadicRight : halfPlane := halfPoint (1/2) (1/2) (by norm_num)

def transferOperator : C(DiskClass,ℝ) →L[ℝ] C(DiskClass,ℝ) :=
  (1/2 : ℝ) • (affineOperator dyadicLeft+affineOperator dyadicRight)

lemma transferOperator_pos (f : C(DiskClass,ℝ)) (hf : 0 ≤ f) : 0 ≤ transferOperator f := by
  intro g
  change 0 ≤ (1/2 : ℝ)*(affineOperator dyadicLeft f g+affineOperator dyadicRight f g)
  exact mul_nonneg (by norm_num) (add_nonneg (affineOperator_pos _ f hf g) (affineOperator_pos _ f hf g))

lemma transferOperator_identityClass (f : C(DiskClass,ℝ)) : transferOperator f identityClass = f identityClass := by
  change (1/2 : ℝ)*(affineOperator dyadicLeft f identityClass+affineOperator dyadicRight f identityClass) = _
  simp only [affineOperator_apply,classWeight_identityClass,reroot_identityClass,one_mul]
  ring

lemma transferOperator_pow_identityClass (n : ℕ) : (transferOperator^n) 1 identityClass = 1 := by
  induction n with
  | zero => rfl
  | succ n ih => simpa only [pow_succ',mul_apply_eq_comp,transferOperator_identityClass] using ih

lemma transferOperator_pow_norm_ge_one (n : ℕ) : 1 ≤ ‖transferOperator^n‖ := by
  have hv := (ContinuousMap.norm_coe_le_norm ((transferOperator^n) 1) identityClass).trans
    ((transferOperator^n).le_opNorm 1)
  simpa only [transferOperator_pow_identityClass,norm_one,mul_one] using hv

lemma diskCayley_norm_sq_identity {z : ℂ} (hz : z ∈ halfPlane) :
    (1-‖diskCayley z‖^2)*‖z+Complex.I‖^2 = 4*z.im := by
  have hd : ‖z+Complex.I‖ ≠ 0 := norm_ne_zero_iff.mpr (cayley_den_ne_zero hz)
  rw [diskCayley,norm_div,div_pow]
  field_simp
  rw [Complex.sq_norm,Complex.sq_norm]
  simp [Complex.normSq_apply]
  ring

lemma classFun_deriv_norm (g : DiskClass) (z : halfPlane) :
    ‖deriv (classFun g) z‖ = 4*‖deriv (diskExtend g.1) (diskCayley z)‖/‖z.val+Complex.I‖^2 := by
  have hd := (hasDerivAt_diskToHalf g.2.1.1 z.2).deriv
  rw [classFun]
  rw [hd,norm_div,norm_mul,norm_neg,norm_pow]
  norm_num

lemma classFun_reciprocal_norm (g : DiskClass) (z : halfPlane) :
    ‖reciprocalDeriv (classFun g) z‖ = ‖z.val+Complex.I‖^2/
      (4*‖deriv (diskExtend g.1) (diskCayley z)‖) := by
  rw [reciprocalDeriv,norm_inv,classFun_deriv_norm,inv_div]

lemma classFun_reciprocal_bound (g : DiskClass) (z : halfPlane) :
    ‖reciprocalDeriv (classFun g) z‖ ≤ ‖z.val+Complex.I‖^4/z.val.im := by
  let r := ‖diskCayley z‖
  let D := ‖z.val+Complex.I‖^2
  let a := ‖deriv (diskExtend g.1) (diskCayley z)‖
  have hr0 : 0 ≤ r := norm_nonneg _
  have hr1 : r < 1 := by simpa [r,disk] using diskCayley_maps z.2
  have hD0 : 0 < D := sq_pos_of_ne_zero (norm_ne_zero_iff.mpr (cayley_den_ne_zero z.2))
  have ha0 : 0 < a := norm_pos_iff.mpr (univalent_deriv_ne_zero Metric.isOpen_ball g.2.1 (diskCayley_maps z.2))
  have he : (1-r^2)*D = 4*z.val.im := diskCayley_norm_sq_identity z.2
  have hlo : (1-r)/(1+r)^3 ≤ a := (schlicht_deriv_bounds g.2 (diskCayley_maps z.2)).1
  have hden : 0 < (1+r)^3 := pow_pos (by linarith) 3
  have h8 : (1+r)^3 ≤ 8 := by nlinarith [sq_nonneg (1-r),sq_nonneg r,mul_nonneg hr0 (sq_nonneg (1-r))]
  have hl : 1-r ≤ 8*a := by
    have hh := (div_le_iff₀ hden).mp hlo
    nlinarith [mul_le_mul_of_nonneg_left h8 ha0.le]
  have hD : 2*z.val.im ≤ (1-r)*D := by
    have hh : (1-r^2) ≤ 2*(1-r) := by nlinarith [sq_nonneg (1-r)]
    nlinarith [mul_le_mul_of_nonneg_right hh hD0.le]
  have hDa : z.val.im ≤ 4*a*D := by
    have hh := mul_le_mul_of_nonneg_right hl hD0.le
    nlinarith
  rw [classFun_reciprocal_norm]
  change D/(4*a) ≤ ‖z.val+Complex.I‖^4/z.val.im
  rw [div_le_div_iff₀ (by positivity : 0 < 4*a) z.2]
  have hh := mul_le_mul_of_nonneg_right hDa hD0.le
  dsimp [D] at *
  nlinarith

lemma classFun_strip_bound (g : DiskClass) {C₀ : ℝ} (hC : 0 ≤ C₀) (z : halfPlane)
    (hx : |z.val.re| ≤ C₀) (hy : z.val.im ≤ 1) :
    ‖reciprocalDeriv (classFun g) z‖ ≤ (C₀^2+4)^2/z.val.im := by
  apply (classFun_reciprocal_bound g z).trans
  apply div_le_div_of_nonneg_right _ z.2.le
  have hD : ‖z.val+Complex.I‖^2 = z.val.re^2+(z.val.im+1)^2 := by
    rw [Complex.sq_norm]
    simp [Complex.normSq_apply]
    ring
  have hxr : z.val.re^2 ≤ C₀^2 := by nlinarith [sq_abs z.val.re,(sq_le_sq₀ (abs_nonneg _) hC).mpr hx]
  have hDle : ‖z.val+Complex.I‖^2 ≤ C₀^2+4 := by
    have hz0 : 0 < z.val.im := z.2
    rw [hD]
    nlinarith [mul_nonneg hz0.le (sub_nonneg.mpr hy)]
  nlinarith [sq_nonneg (‖z.val+Complex.I‖^2-(C₀^2+4))]

lemma classFun_strip_bound_two (g : DiskClass) (z : halfPlane)
    (hx : |z.val.re| ≤ 2) (hy : z.val.im ≤ 1) :
    ‖reciprocalDeriv (classFun g) z‖ ≤ 64/z.val.im := by
  convert classFun_strip_bound g (by norm_num : (0 : ℝ) ≤ 2) z hx hy using 1
  norm_num

end Brennan

end

end OAI