lean/OAI/Geometry/Cannon/Conformal/OpenMapMaxSphere.lean

import Mathlib
import OAI.Geometry.Cannon.Conformal.CutoffBump

namespace OAI

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open MeasureTheory Filter
open scoped Topology

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open Set Metric MeasureTheory Filter Complex
open scoped Topology

namespace CannonPlane

lemma openMap_exists_max_on_sphere {u : ℂ → ℝ} (hu : Continuous u)
    (ho : IsOpenMap u) (c : ℂ) {r : ℝ} (hr : 0 < r) :
    ∃ z ∈ sphere c r, ∀ x ∈ closedBall c r, u x ≤ u z := by
  obtain ⟨z,hz,hmax⟩ := (isCompact_closedBall c r).exists_isMaxOn
    ⟨c,mem_closedBall_self hr.le⟩ hu.continuousOn
  refine ⟨z,?_,hmax⟩
  have hzle : dist z c ≤ r := hz
  apply mem_sphere.mpr
  by_contra he
  have hzlt : dist z c < r := lt_of_le_of_ne hzle he
  have hop : IsOpen (u '' ball c r) := ho _ isOpen_ball
  obtain ⟨ε,hε,hsub⟩ := Metric.isOpen_iff.mp hop (u z) ⟨z,hzlt,rfl⟩
  have hmem : u z+ε/2 ∈ ball (u z) ε := by
    rw [mem_ball,Real.dist_eq]
    have ha : |u z+ε/2-u z|=ε/2 := by rw [add_sub_cancel_left,abs_of_pos (by linarith)]
    rw [ha]; linarith
  obtain ⟨w,hw,hew⟩ := hsub hmem
  have hh : u w ≤ u z := hmax (ball_subset_closedBall hw)
  rw [hew] at hh
  linarith

lemma openMap_exists_min_on_sphere {u : ℂ → ℝ} (hu : Continuous u)
    (ho : IsOpenMap u) (c : ℂ) {r : ℝ} (hr : 0 < r) :
    ∃ z ∈ sphere c r, ∀ x ∈ closedBall c r, u z ≤ u x := by
  obtain ⟨z,hz,hh⟩ := openMap_exists_max_on_sphere hu.neg
    ((Homeomorph.neg ℝ).isOpenMap.comp ho) c hr
  exact ⟨z,hz,fun x hx => neg_le_neg_iff.mp (hh x hx)⟩

lemma homeomorph_disk_oscillation (F : ℂ ≃ₜ ℂ) (c : ℂ) {r L : ℝ}
    (hr : 0 < r) (hb : ∀ x ∈ sphere c r, ∀ y ∈ sphere c r, dist (F x) (F y) ≤ L) :
    ∀ x ∈ closedBall c r, ∀ y ∈ closedBall c r, dist (F x) (F y) ≤ 2*L := by
  obtain ⟨a,ha,hamax⟩ := openMap_exists_max_on_sphere (continuous_re.comp F.continuous)
    (isOpenMap_re.comp F.isOpenMap) c hr
  obtain ⟨b,hb',hbmin⟩ := openMap_exists_min_on_sphere (continuous_re.comp F.continuous)
    (isOpenMap_re.comp F.isOpenMap) c hr
  obtain ⟨d,hd,hdmax⟩ := openMap_exists_max_on_sphere (continuous_im.comp F.continuous)
    (isOpenMap_im.comp F.isOpenMap) c hr
  obtain ⟨e,he,hemin⟩ := openMap_exists_min_on_sphere (continuous_im.comp F.continuous)
    (isOpenMap_im.comp F.isOpenMap) c hr
  intro x hx y hy
  have hre : |(F x-F y).re| ≤ L := by
    have hv := (abs_re_le_norm (F a-F b)).trans (show ‖F a-F b‖ ≤ L by simpa only [dist_eq_norm] using hb a ha b hb')
    rw [abs_le] at hv ⊢
    simp only [sub_re] at hv ⊢
    have h1 := hamax x hx
    have h2 := hamax y hy
    have h3 := hbmin x hx
    have h4 := hbmin y hy
    dsimp only [Function.comp_def] at h1 h2 h3 h4
    constructor <;> linarith [hv.2]
  have him : |(F x-F y).im| ≤ L := by
    have hv := (abs_im_le_norm (F d-F e)).trans (show ‖F d-F e‖ ≤ L by simpa only [dist_eq_norm] using hb d hd e he)
    rw [abs_le] at hv ⊢
    simp only [sub_im] at hv ⊢
    have h1 := hdmax x hx
    have h2 := hdmax y hy
    have h3 := hemin x hx
    have h4 := hemin y hy
    dsimp only [Function.comp_def] at h1 h2 h3 h4
    constructor <;> linarith [hv.2]
  rw [dist_eq_norm]
  linarith [norm_le_abs_re_add_abs_im (F x-F y)]

lemma integral_sq_cauchy {f : ℝ → ℝ} (hf : Continuous f) {a b : ℝ} (hab : a < b) :
    (∫ t in a..b, f t)^2 ≤ (b-a)*(∫ t in a..b, (f t)^2) := by
  let aa := ∫ t in a..b, f t
  let bb := ∫ t in a..b, (f t)^2
  let m := aa/(b-a)
  have hfi := hf.intervalIntegrable (μ := volume) a b
  have hf2 : IntervalIntegrable (fun t => (f t)^2) volume a b :=
    (hf.pow 2).intervalIntegrable a b
  have he : (∫ t in a..b, (f t-m)^2)=bb-2*m*aa+m^2*(b-a) := by
    calc
      _ = ∫ t in a..b, ((f t)^2-2*m*f t+m^2) :=
        intervalIntegral.integral_congr fun _ _ => by ring
      _ = bb-2*m*aa+m^2*(b-a) := by
        rw [intervalIntegral.integral_add (hf2.sub (hfi.const_mul _)) intervalIntegrable_const,
          intervalIntegral.integral_sub hf2 (hfi.const_mul _),intervalIntegral.integral_const_mul,
          intervalIntegral.integral_const]
        simp only [smul_eq_mul,aa,bb]
        ring
  have hp : 0 ≤ ∫ t in a..b, (f t-m)^2 :=
    intervalIntegral.integral_nonneg hab.le (fun _ _ => sq_nonneg _)
  rw [he] at hp
  have ht : 0 < b-a := sub_pos.mpr hab
  have hid : (b-a)*(bb-2*m*aa+m^2*(b-a))=(b-a)*bb-aa^2 := by
    dsimp only [m]
    field_simp
    ring
  have h := mul_nonneg ht.le hp
  rw [hid] at h
  dsimp only [aa,bb] at h
  linarith

lemma circle_derivative_bound {F : ℂ → ℂ} (hf : ContDiff ℝ 1 F) (c : ℂ)
    {r : ℝ} (hr : 0 ≤ r) (θ : ℝ) :
    ‖deriv (fun t => F (circleMap c r t)) θ‖ ≤ r*‖fderiv ℝ F (circleMap c r θ)‖ := by
  have hd := (hf.differentiable (by simp) _).hasFDerivAt.comp_hasDerivAt θ
    (hasDerivAt_circleMap c r θ)
  change ‖deriv (F ∘ circleMap c r) θ‖ ≤ _
  rw [hd.deriv]
  apply le_trans (ContinuousLinearMap.le_opNorm _ _)
  rw [norm_mul,norm_circleMap_zero,Complex.norm_I,abs_of_nonneg hr,mul_one,mul_comm]

lemma circle_oscillation_bound {F : ℂ → ℂ} (hf : ContDiff ℝ 1 F) (c : ℂ)
    {r : ℝ} (hr : 0 < r) :
    ∀ x ∈ sphere c r, ∀ y ∈ sphere c r,
      dist (F x) (F y) ≤ r*(∫ t in (-Real.pi)..Real.pi, ‖fderiv ℝ F (circleMap c r t)‖) := by
  let g : ℝ → ℝ := fun t => r*‖fderiv ℝ F (circleMap c r t)‖
  have hg : Continuous g := continuous_const.mul
    ((hf.continuous_fderiv (by simp)).comp (continuous_circleMap c r)).norm
  have hcircle : circleMap c r '' Ioc (-Real.pi) Real.pi=sphere c r := by
    have h := (periodic_circleMap c r).image_Ioc Real.two_pi_pos (-Real.pi)
    rw [range_circleMap,abs_of_pos hr] at h
    convert h using 2; ring_nf
  intro x hx y hy
  rw [← hcircle] at hx hy
  obtain ⟨a,ha,rfl⟩ := hx
  obtain ⟨b,hb,rfl⟩ := hy
  have hbound (a b : ℝ) (ha : a ∈ Ioc (-Real.pi) Real.pi)
      (hb : b ∈ Ioc (-Real.pi) Real.pi) (hab : a ≤ b) :
      ‖F (circleMap c r b)-F (circleMap c r a)‖ ≤ ∫ t in (-Real.pi)..Real.pi, g t := by
    apply (norm_sub_le_integral_of_norm_deriv_le_of_le hab
      (hf.continuous.comp (continuous_circleMap c r)).continuousOn
      ((hf.differentiable (by simp)).comp (differentiable_circleMap c r)).differentiableOn
      (.of_forall fun t _ => circle_derivative_bound hf c hr.le t) (hg.intervalIntegrable a b)).trans
    exact intervalIntegral.integral_mono_interval ha.1.le hab hb.2
      (.of_forall fun t => mul_nonneg hr.le (norm_nonneg _))
      (hg.intervalIntegrable _ _)
  rw [← intervalIntegral.integral_const_mul]
  change dist (F (circleMap c r a)) (F (circleMap c r b)) ≤ ∫ t in (-Real.pi)..Real.pi, g t
  rcases le_total a b with hab | hba
  · rw [dist_eq_norm, norm_sub_rev]; exact hbound a b ha hb hab
  · rw [dist_eq_norm]; exact hbound b a hb ha hba

lemma homeomorph_circle_energy (F : ℂ ≃ₜ ℂ) (hf : ContDiff ℝ 1 (F : ℂ → ℂ))
    (c : ℂ) {r : ℝ} (hr : 0 < r) :
    ∀ x ∈ closedBall c r, ∀ y ∈ closedBall c r,
      dist (F x) (F y)^2 ≤ 8*Real.pi*r^2*
        (∫ t in (-Real.pi)..Real.pi, ‖fderiv ℝ F (circleMap c r t)‖^2) := by
  intro x hx y hy
  have hosc := homeomorph_disk_oscillation F c hr (circle_oscillation_bound hf c hr) x hx y hy
  have hc : Continuous (fun t => ‖fderiv ℝ F (circleMap c r t)‖) :=
    ((hf.continuous_fderiv (by simp)).comp (continuous_circleMap c r)).norm
  have hcs := integral_sq_cauchy hc (neg_lt_self Real.pi_pos)
  have hn : 0 ≤ ∫ t in (-Real.pi)..Real.pi, ‖fderiv ℝ F (circleMap c r t)‖ :=
    intervalIntegral.integral_nonneg (neg_le_self Real.pi_pos.le) (fun _ _ => norm_nonneg _)
  have hs : dist (F x) (F y)^2 ≤ (2*(r*(∫ t in (-Real.pi)..Real.pi,
      ‖fderiv ℝ F (circleMap c r t)‖)))^2 := sq_le_sq₀ dist_nonneg (by positivity) |>.mpr hosc
  have hm := mul_le_mul_of_nonneg_left hcs (show 0 ≤ 4*r^2 by positivity)
  nlinarith

lemma polar_circle (p : ℝ × ℝ) : Complex.polarCoord.symm p=circleMap 0 p.1 p.2 := by
  rw [Complex.polarCoord_symm_apply,circleMap_zero,Complex.exp_mul_I,Complex.ofReal_cos,Complex.ofReal_sin]

lemma continuous_polar_circle : Continuous (fun p : ℝ × ℝ => circleMap 0 p.1 p.2) := by
  simp only [circleMap]
  fun_prop

lemma disk_polar_integral_zero {g : ℂ → ℝ} (hg : Continuous g) {R : ℝ} (hR : 0 ≤ R) :
    (∫ z in closedBall (0 : ℂ) R, g z) =
      ∫ r in (0 : ℝ)..R, r*(∫ t in (-Real.pi)..Real.pi, g (circleMap 0 r t)) := by
  let A : Set (ℝ × ℝ) := Ioc 0 R ×ˢ Ioo (-Real.pi) Real.pi
  let T : Set (ℝ × ℝ) := Ioi 0 ×ˢ Ioo (-Real.pi) Real.pi
  let q : ℝ × ℝ → ℝ := fun p => p.1*g (circleMap 0 p.1 p.2)
  have hA : MeasurableSet A := measurableSet_Ioc.prod measurableSet_Ioo
  have hT : MeasurableSet T := measurableSet_Ioi.prod measurableSet_Ioo
  have hq : Continuous q := continuous_fst.mul (hg.comp continuous_polar_circle)
  have hi : IntegrableOn q A :=
    (hq.continuousOn.integrableOn_compact (isCompact_Icc.prod isCompact_Icc)).mono_set
      (Set.prod_mono Ioc_subset_Icc_self Ioo_subset_Icc_self)
  have hid (p : ℝ × ℝ) : T.indicator
      (fun p => p.1*(closedBall (0 : ℂ) R).indicator g (Complex.polarCoord.symm p)) p =
      A.indicator q p := by
    by_cases ht : p ∈ T
    · rw [Set.indicator_of_mem ht]
      have hr : 0 < p.1 := ht.1
      have hc : Complex.polarCoord.symm p ∈ closedBall (0 : ℂ) R ↔ p.1 ≤ R := by
        simp only [mem_closedBall,dist_zero_right,Complex.norm_polarCoord_symm,abs_of_pos hr]
      by_cases hle : p.1 ≤ R
      · rw [Set.indicator_of_mem (hc.mpr hle),Set.indicator_of_mem (show p ∈ A from ⟨⟨hr,hle⟩,ht.2⟩)]
        simp only [q,polar_circle]
      · rw [Set.indicator_of_notMem (not_iff_not.mpr hc |>.mpr hle),
          Set.indicator_of_notMem (show p ∉ A from fun hp => hle hp.1.2),mul_zero]
    · rw [Set.indicator_of_notMem ht,Set.indicator_of_notMem
        (show p ∉ A from fun hp => ht ⟨hp.1.1,hp.2⟩)]
  calc
    _ = ∫ z, (closedBall (0 : ℂ) R).indicator g z := (integral_indicator measurableSet_closedBall).symm
    _ = ∫ p in T, p.1*(closedBall (0 : ℂ) R).indicator g (Complex.polarCoord.symm p) := by
      simpa only [T,smul_eq_mul,_root_.polarCoord] using
        (Complex.integral_comp_polarCoord_symm ((closedBall (0 : ℂ) R).indicator g)).symm
    _ = ∫ p, A.indicator q p := by rw [← integral_indicator hT]; exact integral_congr_ae (.of_forall hid)
    _ = ∫ p in A, q p := integral_indicator hA
    _ = ∫ r in Ioc (0 : ℝ) R, ∫ t in Ioo (-Real.pi) Real.pi, r*g (circleMap 0 r t) := by
      rw [Measure.volume_eq_prod] at hi ⊢
      exact setIntegral_prod q hi
    _ = ∫ r in (0 : ℝ)..R, r*(∫ t in (-Real.pi)..Real.pi, g (circleMap 0 r t)) := by
      rw [intervalIntegral.integral_of_le hR]
      apply setIntegral_congr_fun measurableSet_Ioc
      intro r hr
      dsimp only
      rw [integral_const_mul,← integral_Ioc_eq_integral_Ioo,
        intervalIntegral.integral_of_le (neg_le_self Real.pi_pos.le)]

lemma translated_disk_integral (g : ℂ → ℝ) (c : ℂ) (R : ℝ) :
    (∫ z in closedBall (0 : ℂ) R, g (c+z)) = ∫ z in closedBall c R, g z := by
  rw [← integral_indicator measurableSet_closedBall,← integral_indicator measurableSet_closedBall]
  calc
    _ = ∫ z, (closedBall c R).indicator g (c+z) := by
      apply integral_congr_ae
      filter_upwards [] with z
      have he : c+z ∈ closedBall c R ↔ z ∈ closedBall (0 : ℂ) R := by
        simp only [mem_closedBall,dist_eq_norm,add_sub_cancel_left,sub_zero]
      by_cases hz : z ∈ closedBall (0 : ℂ) R
      · rw [Set.indicator_of_mem hz,Set.indicator_of_mem (he.mpr hz)]
      · rw [Set.indicator_of_notMem hz,Set.indicator_of_notMem (not_iff_not.mpr he |>.mpr hz)]
    _ = _ := integral_add_left_eq_self _ c

lemma disk_polar_integral {g : ℂ → ℝ} (hg : Continuous g) (c : ℂ) {R : ℝ} (hR : 0 ≤ R) :
    (∫ z in closedBall c R, g z) =
      ∫ r in (0 : ℝ)..R, r*(∫ t in (-Real.pi)..Real.pi, g (circleMap c r t)) := by
  rw [← translated_disk_integral g c R]
  have hc : Continuous (fun z : ℂ => g (c+z)) := hg.comp (continuous_const.add continuous_id)
  simpa only [circleMap,zero_add] using disk_polar_integral_zero hc hR

lemma continuous_circle_integral {g : ℂ → ℝ} (hg : Continuous g) (c : ℂ) :
    Continuous (fun r : ℝ => ∫ t in (-Real.pi)..Real.pi, g (circleMap c r t)) := by
  apply intervalIntegral.continuous_parametric_intervalIntegral_of_continuous _ continuous_const
  exact hg.comp (by simp only [circleMap]; fun_prop)

lemma approximate_circle_energy (F : ℂ ≃ₜ ℂ) {g : ℂ → ℂ}
    (hg : ContDiff ℝ 1 g) (c : ℂ) {r : ℝ} (hr : 0 < r)
    (x y : ℂ) (hx : x ∈ closedBall c r) (hy : y ∈ closedBall c r)
    (happ : ∀ z ∈ sphere c r, dist (F z) (g z) ≤ dist (F x) (F y)/8) :
    dist (F x) (F y)^2 ≤ 32*Real.pi*r^2*
      (∫ t in (-Real.pi)..Real.pi, ‖fderiv ℝ g (circleMap c r t)‖^2) := by
  let I := ∫ t in (-Real.pi)..Real.pi, ‖fderiv ℝ g (circleMap c r t)‖
  have hb : ∀ a ∈ sphere c r, ∀ b ∈ sphere c r,
      dist (F a) (F b) ≤ r*I+dist (F x) (F y)/4 := by
    intro a ha b hb
    have ho := circle_oscillation_bound hg c hr a ha b hb
    have ht := dist_triangle4 (F a) (g a) (g b) (F b)
    have ha' := happ a ha
    have hb' := happ b hb
    rw [dist_comm (g b) (F b)] at ht
    dsimp only [I] at *
    linarith
  have hosc := homeomorph_disk_oscillation F c hr hb x hx y hy
  have hc : Continuous (fun t => ‖fderiv ℝ g (circleMap c r t)‖) :=
    ((hg.continuous_fderiv (by simp)).comp (continuous_circleMap c r)).norm
  have hcs := integral_sq_cauchy hc (neg_lt_self Real.pi_pos)
  have hn : 0 ≤ I := intervalIntegral.integral_nonneg
    (neg_le_self Real.pi_pos.le) (fun _ _ => norm_nonneg _)
  have hd : dist (F x) (F y) ≤ 4*(r*I) := by linarith
  have hs : dist (F x) (F y)^2 ≤ (4*(r*I))^2 :=
    (sq_le_sq₀ dist_nonneg (by positivity)).mpr hd
  have hm := mul_le_mul_of_nonneg_left hcs (show 0 ≤ 16*r^2 by positivity)
  dsimp only [I] at hs
  nlinarith

lemma approximate_log_energy (F : ℂ ≃ₜ ℂ) {g : ℂ → ℂ}
    (hg : ContDiff ℝ 1 g) (c : ℂ) {s R : ℝ} (hs : 0 < s) (hsR : s ≤ R)
    (x y : ℂ) (hx : x ∈ closedBall c s) (hy : y ∈ closedBall c s)
    (happ : ∀ z ∈ closedBall c R, dist (F z) (g z) ≤ dist (F x) (F y)/8) :
      dist (F x) (F y)^2*Real.log (R/s) ≤
        32*Real.pi*(∫ z in closedBall c R, ‖fderiv ℝ g z‖^2) := by
  let e : ℝ → ℝ := fun r => r*(∫ t in (-Real.pi)..Real.pi, ‖fderiv ℝ g (circleMap c r t)‖^2)
  have hg' : Continuous (fun z => ‖fderiv ℝ g z‖^2) := (hg.continuous_fderiv (by simp)).norm.pow 2
  have he : Continuous e := continuous_id.mul (continuous_circle_integral hg' c)
  have hi : IntervalIntegrable (fun r => dist (F x) (F y)^2/r) volume s R := by
    apply ContinuousOn.intervalIntegrable
    apply continuousOn_const.div continuousOn_id
    intro r hr
    rw [uIcc_of_le hsR] at hr
    exact ne_of_gt (hs.trans_le hr.1)
  have hb : (∫ r in s..R, dist (F x) (F y)^2/r) ≤ ∫ r in s..R, 32*Real.pi*e r := by
    apply intervalIntegral.integral_mono_on hsR hi (he.const_mul _ |>.intervalIntegrable s R)
    intro r hr
    have hrp := hs.trans_le hr.1
    have h := approximate_circle_energy F hg c hrp x y
      (closedBall_subset_closedBall hr.1 hx) (closedBall_subset_closedBall hr.1 hy)
      (fun z hz => happ z (closedBall_subset_closedBall hr.2 (sphere_subset_closedBall hz)))
    apply (div_le_iff₀ hrp).mpr
    dsimp only [e]
    nlinarith
  have hm : (∫ r in s..R, e r) ≤ ∫ r in (0 : ℝ)..R, e r := by
    apply intervalIntegral.integral_mono_interval hs.le hsR le_rfl _ (he.intervalIntegrable _ _)
    apply ae_restrict_of_forall_mem measurableSet_Ioc
    intro r hr
    exact mul_nonneg hr.1.le (intervalIntegral.integral_nonneg (neg_le_self Real.pi_pos.le)
      (fun _ _ => sq_nonneg _))
  rw [show (∫ r in s..R, dist (F x) (F y)^2/r)=dist (F x) (F y)^2*Real.log (R/s) by
    simp only [div_eq_mul_inv]
    rw [intervalIntegral.integral_const_mul,integral_inv_of_pos hs (hs.trans_le hsR)]
    simp only [div_eq_mul_inv]] at hb
  rw [intervalIntegral.integral_const_mul] at hb
  apply hb.trans
  rw [disk_polar_integral hg' c (hs.le.trans hsR)]
  exact mul_le_mul_of_nonneg_left hm (by positivity)
end CannonPlane

namespace CannonMetricQC
open MeasureTheory Set Metric Filter Function ContinuousLinearMap
open scoped Topology Convolution ENNReal NNReal

theorem weak_homeomorph_log_energy (F : ℂ ≃ₜ ℂ) {V W : ℂ → ℂ}
    (hVm : Measurable V) (hWm : Measurable W)
    (hV : LocallyIntegrable V volume) (hW : LocallyIntegrable W volume)
    (hVs : LocallyIntegrable (fun x => ‖V x‖^2) volume)
    (hWs : LocallyIntegrable (fun x => ‖W x‖^2) volume)
    (hv : ∀ φ : SchwartzMap ℂ ℂ, HasCompactSupport (φ : ℂ → ℂ) →
      (∫ z, φ z*V z)= -(∫ z, fderiv ℝ (φ : ℂ → ℂ) z 1*F z))
    (hw : ∀ φ : SchwartzMap ℂ ℂ, HasCompactSupport (φ : ℂ → ℂ) →
      (∫ z, φ z*W z)= -(∫ z, fderiv ℝ (φ : ℂ → ℂ) z Complex.I*F z))
    (c : ℂ) {s R T : ℝ} (hs : 0 < s) (hsR : s ≤ R) (hRT : R < T)
    (x y : ℂ) (hx : x ∈ closedBall c s) (hy : y ∈ closedBall c s) :
    ENNReal.ofReal (dist (F x) (F y)^2*Real.log (R/s)) ≤
      ENNReal.ofReal (64*Real.pi)*(∫⁻ z in closedBall c T,
        ENNReal.ofReal (‖V z‖^2+‖W z‖^2)) := by
  by_cases he : F x=F y
  · simp [he]
  have hd : 0 < dist (F x) (F y)/8 := div_pos (dist_pos.mpr he) (by norm_num)
  have htu := tendstoLocallyUniformly_iff_forall_isCompact.mp
    (mollify_locally_uniform F.continuous) _ (isCompact_closedBall c R)
  have hu := Metric.tendstoUniformlyOn_iff.mp htu _ hd
  have hr := mollifier_radius_tendsto.eventually (gt_mem_nhds (sub_pos.mpr hRT))
  obtain ⟨n,hn,hnr⟩ := (hu.and hr).exists
  have hl := CannonPlane.approximate_log_energy F
    ((mollify_smooth F.continuous.locallyIntegrable n).of_le (by simp)) c hs hsR x y hx hy
    (fun z hz => (hn z hz).le)
  have hE := mollify_opNorm_energy F.continuous hVm hWm hV hW hVs hWs hv hw n c R
  have hm : (∫⁻ z in closedBall c (R+(mollifier n).rOut),
      ENNReal.ofReal (‖V z‖^2+‖W z‖^2)) ≤
      ∫⁻ z in closedBall c T, ENNReal.ofReal (‖V z‖^2+‖W z‖^2) :=
    lintegral_mono_set (closedBall_subset_closedBall (by linarith))
  calc
    _ ≤ ENNReal.ofReal (32*Real.pi)*
        ENNReal.ofReal (∫ z in closedBall c R, ‖fderiv ℝ (mollify F n) z‖^2) := by
      simpa only [ENNReal.ofReal_mul (by positivity : 0 ≤ 32*Real.pi)] using
        ENNReal.ofReal_le_ofReal hl
    _ ≤ ENNReal.ofReal (32*Real.pi)*(2*(∫⁻ z in closedBall c T,
        ENNReal.ofReal (‖V z‖^2+‖W z‖^2))) :=
      mul_le_mul_right (hE.trans (mul_le_mul_right hm 2)) _
    _ = _ := by
      rw [← mul_assoc,← show ENNReal.ofReal (32*Real.pi*2)=ENNReal.ofReal (32*Real.pi)*2 by
        rw [ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_ofNat]]
      congr 2; ring
end CannonMetricQC
noncomputable section
namespace CannonMetricQC
open Set Metric MeasureTheory Filter
open scoped Topology ENNReal NNReal

lemma WeakQS.two {f : ℂ → ℂ} {H : ℝ} (hH : 0 ≤ H) (hf : WeakQS f H)
    (x y z : ℂ) (h : dist x y ≤ 2*dist x z) :
    dist (f x) (f y) ≤ H*(H+1)*dist (f x) (f z) := by
  let u := midpoint ℝ x y
  have hxu : dist x u=dist x y/2 := by simp [u,dist_left_midpoint,div_eq_mul_inv,mul_comm]
  have hux : dist u x=dist x y/2 := by rw [dist_comm,hxu]
  have huy : dist u y=dist x y/2 := by simp [u,dist_midpoint_right,div_eq_mul_inv,mul_comm]
  have h1 := hf x u z (show dist x u ≤ dist x z by rw [hxu]; linarith)
  have h2 := hf u y x (by rw [huy,hux])
  rw [dist_comm (f u) (f x)] at h2
  have h3 := dist_triangle (f x) (f u) (f y)
  have h4 := mul_le_mul_of_nonneg_left h1 (show 0 ≤ H+1 by positivity)
  nlinarith

lemma WeakQS.four {f : ℂ → ℂ} {H : ℝ} (hH : 0 ≤ H) (hf : WeakQS f H)
    (x y z : ℂ) (h : dist x y ≤ 4*dist x z) :
    dist (f x) (f y) ≤ (H*(H+1))^2*dist (f x) (f z) := by
  let u := midpoint ℝ x y
  have hxu : dist x u=dist x y/2 := by simp [u,dist_left_midpoint,div_eq_mul_inv,mul_comm]
  have h1 := hf.two hH x y u (by rw [hxu]; linarith)
  have h2 := hf.two hH x u z (by rw [hxu]; linarith)
  exact h1.trans (by nlinarith [mul_le_mul_of_nonneg_left h2 (show 0 ≤ H*(H+1) by positivity)])

theorem imageArea_upper_four (f : ℂ ≃ₜ ℂ) {H : ℝ} (hH : 0 < H)
    (hf : WeakQS f H) (x y : ℂ) {R : ℝ} (hR : R ≤ 4*dist x y) :
    imageArea f (closedBall x R) ≤ ENNReal.ofReal (Real.pi*(H*(H+1))^4)*
      ENNReal.ofReal (dist (f x) (f y)^2) := by
  have hsub : f '' closedBall x R ⊆ closedBall (f x) ((H*(H+1))^2*dist (f x) (f y)) := by
    rintro w ⟨z,hz,rfl⟩
    rw [mem_closedBall,dist_comm]
    apply hf.four hH.le
    exact (show dist x z ≤ R by simpa only [mem_closedBall,dist_comm] using hz).trans hR
  rw [imageArea_apply f isClosed_closedBall.measurableSet]
  apply (measure_mono (μ:=volume) hsub).trans
  rw [Complex.volume_closedBall]
  have hpi : (NNReal.pi : ℝ≥0∞)=ENNReal.ofReal Real.pi := by
    rw [← NNReal.coe_real_pi,ENNReal.ofReal_coe_nnreal]
  rw [hpi,← ENNReal.ofReal_pow (by positivity),← ENNReal.ofReal_mul (by positivity),
    ← ENNReal.ofReal_mul (by positivity)]
  apply ENNReal.ofReal_le_ofReal
  nlinarith
end CannonMetricQC
noncomputable section
open MeasureTheory Set Metric Filter
open scoped Topology ENNReal

namespace CannonMetricQC

theorem measure_le_of_closedBall_le (μ ν : Measure ℂ) [IsLocallyFiniteMeasure μ]
    [IsLocallyFiniteMeasure ν]
    (h : ∀ x r, 0 < r → μ (closedBall x r) ≤ ν (closedBall x r)) : μ ≤ ν := by
  apply Measure.le_iff.mpr
  intro s _
  apply (Besicovitch.vitaliFamily μ).measure_le_of_frequently_le ν
    Measure.AbsolutelyContinuous.rfl s
  intro x _
  apply (Besicovitch.tendsto_filterAt μ x).frequently
  apply Filter.Eventually.frequently
  filter_upwards [self_mem_nhdsWithin] with r hr
  exact h x r hr

lemma locallyIntegrable_density_locallyFinite {e : ℂ → ℝ}
    (he : LocallyIntegrable e volume) (hp : ∀ x, 0 ≤ e x) :
    IsLocallyFiniteMeasure (volume.withDensity (fun x => ENNReal.ofReal (e x))) where
  finiteAtNhds x := by
    refine ⟨closedBall x 1,closedBall_mem_nhds x (by norm_num),?_⟩
    rw [withDensity_apply _ isClosed_closedBall.measurableSet,
      ← ofReal_integral_eq_lintegral_ofReal (he.integrableOn_isCompact (isCompact_closedBall x 1))
        (.of_forall hp)]
    exact ENNReal.ofReal_lt_top
end CannonMetricQC

namespace CannonMetricQC
open MeasureTheory Set Metric Filter Function ContinuousLinearMap
open scoped Topology Convolution ENNReal NNReal

theorem weakQS_imageArea_le_energy (F : ℂ ≃ₜ ℂ) {H : ℝ} (hH : 0 < H)
    (hF : WeakQS F H) {V W : ℂ → ℂ}
    (hVm : Measurable V) (hWm : Measurable W)
    (hV : LocallyIntegrable V volume) (hW : LocallyIntegrable W volume)
    (hVs : LocallyIntegrable (fun x => ‖V x‖^2) volume)
    (hWs : LocallyIntegrable (fun x => ‖W x‖^2) volume)
    (hv : ∀ φ : SchwartzMap ℂ ℂ, HasCompactSupport (φ : ℂ → ℂ) →
      (∫ z, φ z*V z)= -(∫ z, fderiv ℝ (φ : ℂ → ℂ) z 1*F z))
    (hw : ∀ φ : SchwartzMap ℂ ℂ, HasCompactSupport (φ : ℂ → ℂ) →
      (∫ z, φ z*W z)= -(∫ z, fderiv ℝ (φ : ℂ → ℂ) z Complex.I*F z)) :
    imageArea F ≤ (Real.toNNReal (Real.pi*(H*(H+1))^4*(64*Real.pi/Real.log 2))) •
      volume.withDensity (fun z => ENNReal.ofReal (‖V z‖^2+‖W z‖^2)) := by
  let ν := volume.withDensity (fun z => ENNReal.ofReal (‖V z‖^2+‖W z‖^2))
  let : IsLocallyFiniteMeasure ν := locallyIntegrable_density_locallyFinite
    (hVs.add hWs) (fun z => add_nonneg (sq_nonneg _) (sq_nonneg _))
  apply measure_le_of_closedBall_le
  intro c R hR
  let y : ℂ := c+(R/4:ℝ)
  have hcy : dist c y=R/4 := by
    dsimp [y]
    rw [dist_eq_norm]
    have he : c-(c+(R/4:ℝ))=(-(R/4:ℝ):ℂ) := by push_cast; ring
    rw [he,norm_neg,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (by positivity)]
  have hy : y ∈ closedBall c (R/4) := by rw [mem_closedBall,dist_comm,hcy]
  have hl := weak_homeomorph_log_energy F hVm hWm hV hW hVs hWs hv hw c
    (by positivity : 0 < R/4) (by linarith : R/4 ≤ R/2) (by linarith : R/2 < R)
    c y (mem_closedBall_self (by positivity)) hy
  have heq : R/2/(R/4)=(2:ℝ) := by field_simp; ring
  rw [heq,ENNReal.ofReal_mul (sq_nonneg _)] at hl
  have hp : 0 < Real.log (2:ℝ) := Real.log_pos (by norm_num)
  have hp0 : ENNReal.ofReal (Real.log (2:ℝ)) ≠ 0 := ne_of_gt (ENNReal.ofReal_pos.mpr hp)
  have hle := (ENNReal.le_div_iff_mul_le (Or.inl hp0) (Or.inl ENNReal.ofReal_ne_top)).mpr hl
  have hbound : ENNReal.ofReal (dist (F c) (F y)^2) ≤
      ENNReal.ofReal (64*Real.pi/Real.log 2)*ν (closedBall c R) := by
    dsimp [ν]
    rw [withDensity_apply _ isClosed_closedBall.measurableSet,ENNReal.ofReal_div_of_pos hp]
    simpa only [div_eq_mul_inv,mul_assoc,mul_left_comm,mul_comm] using hle
  have ha := imageArea_upper_four F hH hF c y (show R ≤ 4*dist c y by rw [hcy]; linarith)
  change imageArea F (closedBall c R) ≤
    ENNReal.ofReal (Real.pi*(H*(H+1))^4*(64*Real.pi/Real.log 2))*ν (closedBall c R)
  calc
    _ ≤ ENNReal.ofReal (Real.pi*(H*(H+1))^4)*ENNReal.ofReal (dist (F c) (F y)^2) := ha
    _ ≤ ENNReal.ofReal (Real.pi*(H*(H+1))^4)*
        (ENNReal.ofReal (64*Real.pi/Real.log 2)*ν (closedBall c R)) :=
      mul_le_mul_right hbound _
    _ = _ := by rw [ENNReal.ofReal_mul (show 0 ≤ Real.pi*(H*(H+1))^4 by positivity),mul_assoc]

theorem weakQS_weakW12_imageArea_ac (F : ℂ ≃ₜ ℂ) {H : ℝ} (hH : 0 < H)
    (hF : WeakQS F H) {V W : ℂ → ℂ}
    (hVm : Measurable V) (hWm : Measurable W)
    (hV : LocallyIntegrable V volume) (hW : LocallyIntegrable W volume)
    (hVs : LocallyIntegrable (fun x => ‖V x‖^2) volume)
    (hWs : LocallyIntegrable (fun x => ‖W x‖^2) volume)
    (hv : ∀ φ : SchwartzMap ℂ ℂ, HasCompactSupport (φ : ℂ → ℂ) →
      (∫ z, φ z*V z)= -(∫ z, fderiv ℝ (φ : ℂ → ℂ) z 1*F z))
    (hw : ∀ φ : SchwartzMap ℂ ℂ, HasCompactSupport (φ : ℂ → ℂ) →
      (∫ z, φ z*W z)= -(∫ z, fderiv ℝ (φ : ℂ → ℂ) z Complex.I*F z)) :
    imageArea F ≪ volume := by
  exact (Measure.absolutelyContinuous_of_le
    (weakQS_imageArea_le_energy F hH hF hVm hWm hV hW hVs hWs hv hw)).trans
      (Measure.smul_absolutelyContinuous.trans (withDensity_absolutelyContinuous _ _))
end CannonMetricQC

namespace CannonMetricQC
open MeasureTheory Set Metric Filter
open scoped Topology ENNReal NNReal

lemma weakQS_full_weakW12 (f : ℂ ≃ₜ ℂ) {H : ℝ} (hH : 0 < H) (hf : WeakQS f H) (v : ℂ) :
    LocallyIntegrable (fun x => fderiv ℝ (f : ℂ → ℂ) x v) volume ∧
    LocallyIntegrable (fun x => ‖fderiv ℝ (f : ℂ → ℂ) x v‖^2) volume ∧
    ∀ φ : SchwartzMap ℂ ℂ, HasCompactSupport (φ : ℂ → ℂ) →
      (∫ z, φ z*fderiv ℝ (f : ℂ → ℂ) z v)=
        -(∫ z, fderiv ℝ (φ : ℂ → ℂ) z v*f z) := by
  have hp (K : Set ℂ) (hK : IsCompact K) :
      IntegrableOn (fun x => fderiv ℝ (f : ℂ → ℂ) x v) K volume ∧
      IntegrableOn (fun x => ‖fderiv ℝ (f : ℂ → ℂ) x v‖^2) K volume := by
    obtain ⟨R,hR⟩ := (Metric.isBounded_iff_subset_closedBall (0:ℂ)).mp hK.isBounded
    have hm := (weakQS_local_weakW12 f hH hf R v).1
    let : IsFiniteMeasure (volume.restrict (closedBall (0 : ℂ) R)) :=
      ⟨by simpa using (isCompact_closedBall (0 : ℂ) R).measure_lt_top (μ := volume)⟩
    have hv : IntegrableOn (fun x => fderiv ℝ (f : ℂ → ℂ) x v)
        (closedBall (0 : ℂ) R) volume := MemLp.integrable (by norm_num) hm
    have hs : IntegrableOn (fun x => ‖fderiv ℝ (f : ℂ → ℂ) x v‖^2)
        (closedBall (0 : ℂ) R) volume :=
      (memLp_two_iff_integrable_sq_norm hm.aestronglyMeasurable).mp hm
    exact ⟨hv.mono_set hR, hs.mono_set hR⟩
  refine ⟨locallyIntegrable_iff.mpr (fun K hK => (hp K hK).1),
    locallyIntegrable_iff.mpr (fun K hK => (hp K hK).2),?_⟩
  intro φ hφ
  obtain ⟨R,hR⟩ := (Metric.isBounded_iff_subset_closedBall (0:ℂ)).mp hφ.isBounded
  exact (weakQS_local_weakW12 f hH hf R v).2 φ hR

theorem weakQS_imageArea_ac (f : ℂ ≃ₜ ℂ) {H : ℝ} (hH : 0 < H) (hf : WeakQS f H) :
    imageArea f ≪ volume := by
  obtain ⟨hV,hVs,hv⟩ := weakQS_full_weakW12 f hH hf 1
  obtain ⟨hW,hWs,hw⟩ := weakQS_full_weakW12 f hH hf Complex.I
  exact weakQS_weakW12_imageArea_ac f hH hf
    (measurable_fderiv_apply_const ℝ (f : ℂ → ℂ) 1)
    (measurable_fderiv_apply_const ℝ (f : ℂ → ℂ) Complex.I) hV hW hVs hWs hv hw

theorem weakQS_image_null (f : ℂ ≃ₜ ℂ) {H : ℝ} (hH : 0 < H) (hf : WeakQS f H)
    {S : Set ℂ} (hS : volume S=0) : volume (f '' S)=0 := by
  have ht := weakQS_imageArea_ac f hH hf
  have hz : imageArea f (toMeasurable volume S)=0 := ht (by simpa only [measure_toMeasurable] using hS)
  rw [imageArea_apply f (measurableSet_toMeasurable _ _)] at hz
  exact measure_mono_null (image_mono (subset_toMeasurable volume S)) hz
end CannonMetricQC

namespace CannonMetricQC
open MeasureTheory Filter
open scoped Topology

theorem weakQS_quasiMeasurePreserving (f : ℂ ≃ₜ ℂ) {H : ℝ}
    (hH : 0 < H) (hf : WeakQS f.symm H) :
    Measure.QuasiMeasurePreserving (f : ℂ → ℂ) volume volume := by
  refine ⟨f.continuous.measurable,?_⟩
  simpa only [imageArea,Homeomorph.symm_symm] using weakQS_imageArea_ac f.symm hH hf

theorem weakQS_ae_bijective_fderiv (f : ℂ ≃ₜ ℂ) {H J : ℝ}
    (hH : 0 < H) (hJ : 0 < J) (hf : WeakQS f H) (hi : WeakQS f.symm J) :
    ∀ᵐ x : ℂ, Function.Bijective (fderiv ℝ (f : ℂ → ℂ) x) := by
  have hm := weakQS_quasiMeasurePreserving f hJ hi
  filter_upwards [weakQS_ae_differentiable f hH hf,
    hm.ae (weakQS_ae_differentiable f.symm hJ hi)] with x hx hy
  have hc := hy.hasFDerivAt.comp x hx.hasFDerivAt
  have hid : (f.symm : ℂ → ℂ) ∘ (f : ℂ → ℂ)=id := by funext z; simp
  rw [hid] at hc
  have hh := hc.unique (hasFDerivAt_id x)
  have hl : Function.LeftInverse (fderiv ℝ (f.symm : ℂ → ℂ) (f x))
      (fderiv ℝ (f : ℂ → ℂ) x) := by
    intro z
    exact congrArg (fun A : ℂ →L[ℝ] ℂ => A z) hh
  exact ⟨hl.injective,LinearMap.injective_iff_surjective.mp hl.injective⟩

theorem weakQS_ae_fderiv_comp (f g : ℂ ≃ₜ ℂ) {H J K : ℝ}
    (hH : 0 < H) (hJ : 0 < J) (hK : 0 < K)
    (hf : WeakQS f H) (hg : WeakQS g J) (hgi : WeakQS g.symm K) :
    ∀ᵐ x : ℂ, fderiv ℝ (fun z => f (g z)) x =
      (fderiv ℝ (f : ℂ → ℂ) (g x)).comp (fderiv ℝ (g : ℂ → ℂ) x) := by
  filter_upwards [weakQS_ae_differentiable g hJ hg,
    (weakQS_quasiMeasurePreserving g hK hgi).ae (weakQS_ae_differentiable f hH hf)]
      with x hx hy
  exact (hy.hasFDerivAt.comp x hx.hasFDerivAt).fderiv
end CannonMetricQC

end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end

end OAI