lean/OAI/Geometry/ProjectionBodies/SignSpectrum.lean

import OAI.Geometry.ProjectionBodies.CosineSpectrum

namespace OAI

universe uE

noncomputable section
open Set MeasureTheory Metric Filter Topology Function
open scoped ENNReal RealInnerProductSpace
namespace PettyProjection.Spherical

def signFactor (n j : ℕ) (a : ℝ) : ℝ :=
  higherFactor n j 1*(1+a*eigenvalue n (2*j))

lemma signFactor_abs_step {n j : ℕ} (hn : 2 ≤ n) (hj : 1 ≤ j) {a : ℝ} (ha : 0 ≤ a) :
    |signFactor n (j+1) a| ≤ |signFactor n j a| := by
  have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn
  have hj' : (1 : ℝ) ≤ j := by exact_mod_cast hj
  have hden : 0 < 1+(n : ℝ)+2*j := by linarith
  have he : 0 ≤ eigenvalue n (2*j) := by
    unfold eigenvalue
    push_cast
    apply mul_nonneg (by positivity)
    linarith
  have he' : 0 ≤ eigenvalue n (2*(j+1)) := by
    unfold eigenvalue
    push_cast
    apply mul_nonneg (by positivity)
    linarith
  have hb : (2*(j : ℝ)-1)*(1+a*eigenvalue n (2*(j+1))) ≤
      (1+(n : ℝ)+2*j)*(1+a*eigenvalue n (2*j)) := by
    have hpos : 0 ≤ a*(((n : ℝ)-2)*(2*j)*(2*j+n)+2*n) := by positivity
    simp only [eigenvalue, Nat.cast_mul, Nat.cast_ofNat, Nat.cast_add, Nat.cast_one]
    nlinarith
  simp only [signFactor, higherFactor_succ, abs_mul]
  rw [abs_div, abs_of_pos hden, abs_of_nonpos (by linarith : 1-2*(j : ℝ) ≤ 0), neg_sub,
    abs_of_nonneg (by positivity : 0 ≤ 1+a*eigenvalue n (2*(j+1))),
    abs_of_nonneg (by positivity : 0 ≤ 1+a*eigenvalue n (2*j))]
  rw [mul_assoc]
  apply mul_le_mul_of_nonneg_left _ (abs_nonneg _)
  rw [div_mul_eq_mul_div, div_le_iff₀ hden]
  simpa only [mul_comm] using hb

lemma signFactor_abs_le_four {n j : ℕ} (hn : 2 ≤ n) (hj : 2 ≤ j) {a : ℝ} (ha : 0 ≤ a) :
    |signFactor n j a| ≤ (1+a*(4*(n+2)))/((n+1)*(n+3)) := by
  have hstep : |signFactor n j a| ≤ |signFactor n 2 a| := by
    induction j, hj using Nat.le_induction with
    | base => exact le_rfl
    | succ j hj ih => exact (signFactor_abs_step hn (by omega) ha).trans ih
  have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn
  have he : signFactor n 2 a = -(1+a*(4*(n+2)))/((n+1)*(n+3)) := by
    norm_num [signFactor, higherFactor, Finset.prod_range_succ, eigenvalue]
    field_simp
    ring
  rw [he, abs_of_nonpos (by exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (by positivity)) (by positivity) : -(1+a*(4*(n+2)))/((n+1)*(n+3)) ≤ 0)] at hstep
  simpa only [neg_div, neg_neg] using hstep

end PettyProjection.Spherical
end

noncomputable section
open Set MeasureTheory Metric Filter Topology MvPolynomial
open scoped ENNReal RealInnerProductSpace
namespace PettyProjection.Spherical

lemma homogeneous_two_mem_harmonic_sup {n : ℕ} (hn : 2 ≤ n) [NeZero n]
    {p : MvPolynomial (Fin n) ℝ} (hp : p.IsHomogeneous 2) :
    polynomialL2 n p ∈ harmonicSpace n 0 ⊔ harmonicSpace n 2 := by
  have hs := homogeneous_mem_fischerSpan hn 2 hp
  clear hp
  induction hs using Submodule.span_induction with
  | mem x hx =>
    obtain ⟨j,l,p,hd,hp,hΔ,rfl⟩ := hx
    have hr : polynomialL2 n (radiusPolynomial n ^ j * p) = polynomialL2 n p := by
      change toH n (polynomialRestriction n _) = toH n (polynomialRestriction n p)
      rw [map_mul, map_pow, polynomialRestriction_radius, one_pow, one_mul]
    rw [hr]
    have hl : l = 0 ∨ l = 2 := by omega
    rcases hl with rfl | rfl
    · exact Submodule.mem_sup_left ⟨p,⟨hp,hΔ⟩,rfl⟩
    · exact Submodule.mem_sup_right ⟨p,⟨hp,hΔ⟩,rfl⟩
  | zero => simp
  | add p q _ _ hp hq => simpa only [map_add] using Submodule.add_mem _ hp hq
  | smul c p _ hp => simpa only [map_smul] using Submodule.smul_mem _ c hp

lemma lowSpace_eq {n : ℕ} (hn : 2 ≤ n) [NeZero n] :
    lowSpace n = harmonicSpace n 0 ⊔ harmonicSpace n 2 := by
  apply le_antisymm
  · apply Submodule.span_le.mpr
    rintro _ ⟨⟨i,j⟩,rfl⟩
    change toH n (coordinateProduct n i j) ∈ _
    rw [← polynomialL2_X_mul_X]
    exact homogeneous_two_mem_harmonic_sup hn ((isHomogeneous_X ℝ i).mul (isHomogeneous_X ℝ j))
  · apply sup_le
    · intro v hv
      obtain ⟨c,rfl⟩ := harmonicSpace_zero_constant hv
      exact constant_mem_lowSpace n c
    · exact harmonicSpace_two_le_lowSpace n

lemma harmonic_inner_ne {n : ℕ} (hn : 2 ≤ n) [NeZero n] {d e : ℕ} (hde : d ≠ e)
    {v w : H n} (hv : v ∈ harmonicSpace n d) (hw : w ∈ harmonicSpace n e) :
    ⟪v,w⟫ = 0 := by
  exact harmonicSpace_orthogonal hn hde ⟨v,hv⟩ ⟨w,hw⟩

lemma lowSpace_projection {n : ℕ} (hn : 2 ≤ n) [NeZero n] (v : H n) :
    (lowSpace n).starProjection v = (harmonicSpace n 0).starProjection v +
      (harmonicSpace n 2).starProjection v := by
  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero
  · rw [lowSpace_eq hn]
    exact Submodule.add_mem _ (Submodule.mem_sup_left ((harmonicSpace n 0).starProjection_apply_mem v))
      (Submodule.mem_sup_right ((harmonicSpace n 2).starProjection_apply_mem v))
  · intro w hw
    rw [lowSpace_eq hn, Submodule.mem_sup] at hw
    obtain ⟨w0,hw0,w2,hw2,rfl⟩ := hw
    rw [inner_add_right]
    have h0 : ⟪(harmonicSpace n 2).starProjection v,w0⟫ = 0 :=
      harmonic_inner_ne hn (by decide) ((harmonicSpace n 2).starProjection_apply_mem v) hw0
    have h2 : ⟪(harmonicSpace n 0).starProjection v,w2⟫ = 0 :=
      harmonic_inner_ne hn (by decide) ((harmonicSpace n 0).starProjection_apply_mem v) hw2
    simp only [inner_sub_left, inner_add_left, h0, h2, add_zero, zero_add]
    rw [Submodule.inner_starProjection_left_eq_right,
      Submodule.starProjection_eq_self_iff.mpr hw0,
      Submodule.inner_starProjection_left_eq_right,
      Submodule.starProjection_eq_self_iff.mpr hw2]
    ring

lemma Q_eq_sub_projections {n : ℕ} (hn : 2 ≤ n) [NeZero n] (v : H n) :
    Q n v = v - (harmonicSpace n 0).starProjection v - (harmonicSpace n 2).starProjection v := by
  rw [Q_eq_sub, lowSpace_projection hn, sub_add_eq_sub_sub]

lemma harmonic_projection_of_mem_ne {n : ℕ} (hn : 2 ≤ n) [NeZero n] {d e : ℕ} (hde : d ≠ e)
    {v : H n} (hv : v ∈ harmonicSpace n e) : (harmonicSpace n d).starProjection v = 0 := by
  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero (Submodule.zero_mem _)
  intro w hw
  rw [sub_zero]
  exact harmonic_inner_ne hn hde.symm hv hw

lemma harmonic_projection_Q {n d : ℕ} (hn : 2 ≤ n) [NeZero n] (v : H n) :
    (harmonicSpace n d).starProjection (Q n v) =
      if d = 0 ∨ d = 2 then 0 else (harmonicSpace n d).starProjection v := by
  rw [Q_eq_sub_projections hn, map_sub, map_sub]
  by_cases h0 : d = 0
  · subst d
    rw [ite_eq_left (Or.inl rfl), Submodule.starProjection_eq_self_iff.mpr ((harmonicSpace n _).starProjection_apply_mem v),
      harmonic_projection_of_mem_ne hn (by decide) ((harmonicSpace n 2).starProjection_apply_mem v)]
    simp
  · by_cases h2 : d = 2
    · subst d
      rw [ite_eq_left (Or.inr rfl), Submodule.starProjection_eq_self_iff.mpr ((harmonicSpace n _).starProjection_apply_mem v),
        harmonic_projection_of_mem_ne hn (by decide) ((harmonicSpace n 0).starProjection_apply_mem v)]
      simp
    · rw [ite_eq_right (not_or.mpr ⟨h0,h2⟩),
        harmonic_projection_of_mem_ne hn h0 ((harmonicSpace n 0).starProjection_apply_mem v),
        harmonic_projection_of_mem_ne hn h2 ((harmonicSpace n 2).starProjection_apply_mem v)]
      simp

end PettyProjection.Spherical
end

noncomputable section
open MeasureTheory Set Filter Topology
open scoped ENNReal RealInnerProductSpace
namespace PettyProjection
variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
variable {F : ℕ → Submodule ℝ E} [∀ i, CompleteSpace (F i)]

lemma hilbertSum_norm_le_of_projection_bound
    (h : IsHilbertSum ℝ (fun i => F i) (fun i => (F i).subtypeₗᵢ))
    (x y : E) {c : ℝ} (hc : 0 ≤ c)
    (hp : ∀ i, ‖(F i).starProjection y‖ ≤ c*‖(F i).starProjection x‖) :
    ‖y‖ ≤ c*‖x‖ := by
  have hsq : ‖y‖^2 ≤ c^2*‖x‖^2 := by
    apply hasSum_le _ (hilbertSum_hasSum_projection_norm_sq h y)
      (HasSum.mul_left (c^2) (hilbertSum_hasSum_projection_norm_sq h x))
    intro i
    have h := sq_le_sq₀ (norm_nonneg ((F i).starProjection y))
      (mul_nonneg hc (norm_nonneg ((F i).starProjection x))) |>.mpr (hp i)
    simpa only [mul_pow] using h
  nlinarith [norm_nonneg y, mul_nonneg hc (norm_nonneg x)]

omit [CompleteSpace E] in
lemma inner_projection_both (K : Submodule ℝ E) [CompleteSpace K] (x y : E) :
    ⟪K.starProjection x,K.starProjection y⟫ = ⟪x,K.starProjection y⟫ := by
  rw [Submodule.inner_starProjection_left_eq_right,
    Submodule.starProjection_eq_self_iff.mpr (K.starProjection_apply_mem y)]

lemma hilbertSum_hasSum_inner
    (h : IsHilbertSum ℝ (fun i => F i) (fun i => (F i).subtypeₗᵢ)) (x y : E) :
    HasSum (fun i => ⟪(F i).starProjection x,(F i).starProjection y⟫) ⟪x,y⟫ := by
  simp only [inner_projection_both]
  exact (innerSL ℝ x).hasSum (hilbertSum_hasSum_projection h y)

lemma hilbertSum_multiplier_selfadj
    (h : IsHilbertSum ℝ (fun i => F i) (fun i => (F i).subtypeₗᵢ))
    (x y tx ty : E) (c : ℕ → ℝ)
    (hx : ∀ i, (F i).starProjection tx = c i • (F i).starProjection x)
    (hy : ∀ i, (F i).starProjection ty = c i • (F i).starProjection y) :
    ⟪x,ty⟫ = ⟪tx,y⟫ := by
  have h₁ := hilbertSum_hasSum_inner h x ty
  have h₂ := hilbertSum_hasSum_inner h tx y
  simp only [hx, hy, real_inner_smul_left, real_inner_smul_right] at h₁ h₂
  exact h₁.unique h₂

end PettyProjection
end

noncomputable section
open Set MeasureTheory Metric Filter Topology Function
open scoped ENNReal RealInnerProductSpace
namespace PettyProjection.Spherical

lemma casimir_continuous {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) :
    Continuous (casimir f) := by
  unfold casimir
  exact continuous_const.mul (continuous_finsetSum _ fun i _ =>
    continuous_finsetSum _ fun j _ => continuous_rotDeriv _ (contDiff_rotDeriv _ hf))

def casimirMap {n : ℕ} (f : Space n → ℝ) (hf : ContDiff ℝ 2 f) : C(Sphere n, ℝ) :=
  restrictContinuous (casimir f) (casimir_continuous hf)

lemma casimir_projection {n d : ℕ} [NeZero n] {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) :
    (harmonicSpace n d).starProjection (toH n (casimirMap f hf)) =
      -eigenvalue n d • (harmonicSpace n d).starProjection (toH n (restrictContinuous f hf.continuous)) := by
  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero
    ((harmonicSpace n d).smul_mem _ ((harmonicSpace n d).starProjection_apply_mem _))
  rintro v ⟨p,hp,rfl⟩
  rw [inner_sub_left, real_inner_smul_left, Submodule.inner_starProjection_left_eq_right,
    Submodule.starProjection_eq_self_iff.mpr (show polynomialL2 n p ∈ harmonicSpace n d from ⟨p,hp,rfl⟩)]
  change ⟪toH n (casimirMap f hf),toH n (polynomialRestriction n p)⟫ -
    (-eigenvalue n d)*⟪toH n (restrictContinuous f hf.continuous),toH n (polynomialRestriction n p)⟫ = 0
  rw [inner_toH, inner_toH]
  change mean (fun x : Sphere n => casimir f x*polynomialFunction p x) -
    (-eigenvalue n d)*mean (fun x : Sphere n => f x*polynomialFunction p x) = 0
  simp_rw [mul_comm (casimir f _)]
  rw [mean_casimir_comm ((polynomialFunction_contDiff p).of_le le_top) hf]
  simp_rw [casimir_harmonic hp.1 hp.2, mul_left_comm (f _) (-eigenvalue n d)]
  rw [mean, integral_const_mul]
  exact sub_self _

def signOperator {n : ℕ} [NeZero n] (a : ℝ) (f : Space n → ℝ) (hf : ContDiff ℝ 2 f) :
    C(Sphere n, ℝ) := cosineMap (restrictContinuous f hf.continuous - a • casimirMap f hf)

lemma signOperator_projection_even {n j : ℕ} (hn : 2 ≤ n) [NeZero n]
    (a : ℝ) {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) :
    (harmonicSpace n (2*j)).starProjection (toH n (signOperator a f hf)) =
      signFactor n j a • (harmonicSpace n (2*j)).starProjection (toH n (restrictContinuous f hf.continuous)) := by
  rw [signOperator, cosine_projection_even hn, map_sub, map_smul, map_sub, map_smul,
    casimir_projection hf, smul_smul, smul_sub, smul_smul, ← sub_smul]
  congr 1
  unfold signFactor
  ring

lemma signOperator_projection_odd {n d : ℕ} [NeZero n]
    (a : ℝ) {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) (hd : Odd d) :
    (harmonicSpace n d).starProjection (toH n (signOperator a f hf)) = 0 :=
  cosine_projection_odd _ hd

lemma norm_Q_signOperator_le {n : ℕ} (hn : 2 ≤ n) [NeZero n]
    {a : ℝ} (ha : 0 ≤ a) {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) :
    ‖Q n (toH n (signOperator a f hf))‖ ≤
      ((1+a*(4*(n+2)))/((n+1)*(n+3)))*‖Q n (toH n (restrictContinuous f hf.continuous))‖ := by
  apply hilbertSum_norm_le_of_projection_bound (harmonicSpace_isHilbertSum hn) _ _ (by positivity)
  intro d
  rw [harmonic_projection_Q hn, harmonic_projection_Q hn]
  split_ifs with hd
  · simp
  · by_cases he : Even d
    · obtain ⟨j,hj⟩ := he
      have hde : d = 2*j := by omega
      rw [hde]
      have hj2 : 2 ≤ j := by omega
      rw [signOperator_projection_even hn, norm_smul, Real.norm_eq_abs]
      exact mul_le_mul_of_nonneg_right (signFactor_abs_le_four hn hj2 ha) (norm_nonneg _)
    · rw [signOperator_projection_odd a hf (Nat.not_even_iff_odd.mp he), norm_zero]
      positivity

lemma signOperator_selfadj {n : ℕ} (hn : 2 ≤ n) [NeZero n] (a : ℝ)
    {f g : Space n → ℝ} (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) :
    mean (fun x : Sphere n => f x*signOperator a g hg x) =
      mean (fun x : Sphere n => signOperator a f hf x*g x) := by
  change mean (fun x => restrictContinuous f hf.continuous x*signOperator a g hg x) =
    mean (fun x => signOperator a f hf x*restrictContinuous g hg.continuous x)
  rw [← inner_toH n (restrictContinuous f hf.continuous) (signOperator a g hg),
    ← inner_toH n (signOperator a f hf) (restrictContinuous g hg.continuous)]
  let c : ℕ → ℝ := fun d => if Even d then signFactor n (d/2) a else 0
  apply hilbertSum_multiplier_selfadj (harmonicSpace_isHilbertSum hn) _ _ _ _ c
  all_goals
    intro d
    by_cases he : Even d
    · obtain ⟨j,hj⟩ := he
      have hd : d = 2*j := by omega
      rw [hd]
      simp only [c, even_two_mul, ↓reduceIte, Nat.mul_div_cancel_left _ (by decide : 0 < 2)]
      exact signOperator_projection_even hn a _
    · simp only [c, he, ↓reduceIte, zero_smul]
      exact signOperator_projection_odd a _ (Nat.not_even_iff_odd.mp he)

end PettyProjection.Spherical
end

noncomputable section
open Set MeasureTheory Metric Filter Topology Function
open scoped ENNReal RealInnerProductSpace
namespace PettyProjection.Spherical

lemma cosine_sub_smul {n : ℕ} [NeZero n] {f g : Sphere n → ℝ}
    (hf : Continuous f) (hg : Continuous g) (a : ℝ) (u : Sphere n) :
    cosine (fun x => f x-a*g x) u = cosine f u-a*cosine g u := by
  unfold cosine mean
  simp_rw [mul_sub, mul_left_comm _ a]
  rw [integral_sub (continuous_integrable (by fun_prop)) (continuous_integrable (by fun_prop)),
    integral_const_mul]
  ring

lemma signOperator_apply {n : ℕ} [NeZero n] (a : ℝ) {f : Space n → ℝ}
    (hf : ContDiff ℝ 2 f) (u : Sphere n) :
    signOperator a f hf u = cosine (fun x : Sphere n => f x) u -
      a*cosine (fun x : Sphere n => casimir f x) u := by
  exact cosine_sub_smul (hf.continuous.comp continuous_subtype_val)
    ((casimir_continuous hf).comp continuous_subtype_val) a u

lemma casimir_const {n : ℕ} (c : ℝ) : casimir (fun _ : Space n => c) = 0 := by
  have hr (X : Space n →L[ℝ] Space n) : rotDeriv X (fun _ => c) = 0 := by
    ext x
    simp [rotDeriv]
  ext x
  simp [casimir, hr, rotDeriv]

lemma mean_casimir_zero {n : ℕ} [NeZero n] {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) :
    mean (fun x : Sphere n => casimir f x) = 0 := by
  have h := mean_casimir_comm (f := fun _ => 1) contDiff_const hf
  simpa only [one_mul, casimir_const, Pi.zero_apply, mul_zero, mean, integral_zero] using h

lemma mean_signOperator {n : ℕ} [NeZero n] (a : ℝ) {f : Space n → ℝ}
    (hf : ContDiff ℝ 2 f) : mean (signOperator a f hf) = mean (fun x : Sphere n => f x) := by
  rw [signOperator, show (cosineMap (restrictContinuous f hf.continuous - a • casimirMap f hf) : Sphere n → ℝ) =
      cosine (restrictContinuous f hf.continuous - a • casimirMap f hf) from rfl,
    mean_cosine (ContinuousMap.continuous _)]
  change mean (fun x : Sphere n => f x-a*casimir f x) = _
  unfold mean
  rw [integral_sub (f := fun x : Sphere n => f x) (g := fun x : Sphere n => a*casimir f x)
    (continuous_integrable (hf.continuous.comp continuous_subtype_val))
    (continuous_integrable (continuous_const.mul ((casimir_continuous hf).comp continuous_subtype_val))),
    integral_const_mul]
  change mean (fun x : Sphere n => f x) - a*mean (fun x : Sphere n => casimir f x) = _
  rw [mean_casimir_zero hf, mul_zero, sub_zero]
  rfl

lemma mean_signOperator_expand {n : ℕ} [NeZero n] (a : ℝ) {f g : Space n → ℝ}
    (hf : Continuous f) (hg : ContDiff ℝ 2 g) :
    mean (fun x : Sphere n => f x*signOperator a g hg x) =
      mean (fun x : Sphere n => f x*cosine (fun y : Sphere n => g y) x) -
        a*mean (fun x : Sphere n => f x*cosine (fun y : Sphere n => casimir g y) x) := by
  simp_rw [signOperator_apply, mul_sub, mul_left_comm (f _) a]
  exact (integral_sub
    (continuous_integrable ((hf.comp continuous_subtype_val).mul (cosine_continuous (hg.continuous.comp continuous_subtype_val))))
    (continuous_integrable (continuous_const.mul ((hf.comp continuous_subtype_val).mul
      (cosine_continuous ((casimir_continuous hg).comp continuous_subtype_val)))))).trans
    (by
      change _ - (∫ x : Sphere n, a * (f x * cosine (fun y : Sphere n => casimir g y) x) ∂sigma n) = _
      rw [integral_const_mul]
      rfl)

lemma cosine_casimir_selfadj {n : ℕ} (hn : 2 ≤ n) [NeZero n]
    {f g : Space n → ℝ} (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) :
    mean (fun x : Sphere n => f x*cosine (fun y : Sphere n => casimir g y) x) =
      mean (fun x : Sphere n => g x*cosine (fun y : Sphere n => casimir f y) x) := by
  have h := signOperator_selfadj hn 1 hf hg
  simp_rw [mul_comm (signOperator 1 f hf _)] at h
  rw [mean_signOperator_expand 1 hf.continuous hg, mean_signOperator_expand 1 hg.continuous hf] at h
  have hc := cosine_selfadj (hf.continuous.comp continuous_subtype_val)
    (hg.continuous.comp continuous_subtype_val)
  simp_rw [mul_comm (cosine _ _)] at hc
  change (mean (fun x : Sphere n => f x * cosine (fun y : Sphere n => g y) x)) =
    mean (fun x : Sphere n => g x * cosine (fun y : Sphere n => f y) x) at hc
  linarith

end PettyProjection.Spherical
end

noncomputable section
open Set MeasureTheory Metric Filter Topology Function
open scoped ENNReal RealInnerProductSpace
namespace PettyProjection.Spherical

lemma inner_Q_Q {n : ℕ} [NeZero n] (x y : H n) :
    ⟪Q n x,Q n y⟫ = ⟪Q n x,y⟫ := by
  change ⟪(lowSpace n)ᗮ.starProjection x,(lowSpace n)ᗮ.starProjection y⟫ = _
  rw [Submodule.inner_starProjection_left_eq_right,
    Submodule.starProjection_eq_self_iff.mpr ((lowSpace n)ᗮ.starProjection_apply_mem y)]
  exact (Submodule.inner_starProjection_left_eq_right _ _ _).symm

lemma inner_Q_Q_of_degree_two_zero {n : ℕ} (hn : 2 ≤ n) [NeZero n]
    (f g : C(Sphere n, ℝ)) (h2 : (harmonicSpace n 2).starProjection (toH n f) = 0) :
    ⟪Q n (toH n f),Q n (toH n g)⟫ = mean (fun u => f u*g u) - mean f*mean g := by
  rw [inner_Q_Q, Q_eq_sub_projections hn, h2, sub_zero, inner_sub_left,
    harmonicSpace_zero_projection, inner_toH, inner_toH]
  change _ - mean (fun u => mean f*g u) = _
  congr 1
  exact integral_const_mul (mean f) g

lemma sign_spectral_estimate {n : ℕ} (hn : 2 ≤ n) [NeZero n]
    {a : ℝ} (ha : 0 ≤ a) (f : C(Sphere n, ℝ)) {g : Space n → ℝ}
    (hg : ContDiff ℝ 2 g) (h2 : (harmonicSpace n 2).starProjection (toH n f) = 0) :
    mean f*mean (fun x : Sphere n => g x) -
      ((1+a*(4*(n+2)))/((n+1)*(n+3)))*‖Q n (toH n f)‖*
        ‖Q n (toH n (restrictContinuous g hg.continuous))‖ ≤
      mean (fun x => f x*signOperator a g hg x) := by
  have hi := inner_Q_Q_of_degree_two_zero hn f (signOperator a g hg) h2
  rw [mean_signOperator] at hi
  have hcs := neg_le_of_abs_le (abs_real_inner_le_norm
    (Q n (toH n f)) (Q n (toH n (signOperator a g hg))))
  have hb := mul_le_mul_of_nonneg_left (norm_Q_signOperator_le hn ha hg)
    (norm_nonneg (Q n (toH n f)))
  rw [hi] at hcs
  nlinarith

end PettyProjection.Spherical
end

noncomputable section
open Set MeasureTheory Metric Filter Topology Function
open scoped ENNReal RealInnerProductSpace Gradient
namespace PettyProjection.Spherical

def radialField {n : ℕ} (f : Space n → ℝ) (u : Sphere n) : Space n := f u • (u : Space n)
def tangentField {n : ℕ} (f : Space n → ℝ) (u : Sphere n) : Space n := sphereGradient f u

def signField {n : ℕ} (k : ℝ) (f : Space n → ℝ) (u : Sphere n) : Space n :=
  radialField f u + k⁻¹ • tangentField f u

lemma radialField_continuous {n : ℕ} {f : Space n → ℝ} (hf : Continuous f) :
    Continuous (radialField f) := (hf.comp continuous_subtype_val).smul continuous_subtype_val

lemma tangentField_continuous {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) :
    Continuous (tangentField f) := (sphereGradient_continuous hf).comp continuous_subtype_val

lemma signedPair_radial_radial {n : ℕ} [NeZero n] {f g : Space n → ℝ}
    (hf : Continuous f) (hg : Continuous g) :
    signedPair (radialField f) (radialField g) = cosineConstant n *
      mean (fun u : Sphere n => f u*cosine (fun v : Sphere n => g v) u) := by
  rw [signedPair_eq (radialField_continuous hf) (radialField_continuous hg)]
  have hi (u v : Sphere n) :
      Real.sign ⟪(u : Space n),(v : Space n)⟫ * ⟪radialField f u,radialField g v⟫ =
        f u*(|⟪(u : Space n),(v : Space n)⟫| * g v) := by
    simp only [radialField, real_inner_smul_left, real_inner_smul_right]
    calc
      _ = (Real.sign ⟪(u : Space n),(v : Space n)⟫ * ⟪(u : Space n),(v : Space n)⟫)*f u*g v := by ring
      _ = _ := by rw [sign_mul_self]; ring
  simp_rw [hi, mean, integral_const_mul]
  simp only [cosine, mean, ← mul_div_assoc, integral_div]
  field_simp [(cosineConstant_pos n).ne']

lemma signedPair_tangent_radial {n : ℕ} [NeZero n] {f g : Space n → ℝ}
    (hf : ContDiff ℝ 2 f) (hg : Continuous g) :
    signedPair (tangentField f) (radialField g) = -cosineConstant n *
      mean (fun u : Sphere n => g u*cosine (fun v : Sphere n => casimir f v) u) := by
  rw [signedPair_eq_swap (tangentField_continuous (hf.of_le (by norm_num))) (radialField_continuous hg)]
  have hi (u v : Sphere n) :
      Real.sign ⟪(u : Space n),(v : Space n)⟫ * ⟪tangentField f u,radialField g v⟫ =
        g v*(Real.sign ⟪(v : Space n),(u : Space n)⟫*⟪sphereGradient f u,(v : Space n)⟫) := by
    simp only [radialField, tangentField, real_inner_smul_right]
    rw [real_inner_comm (u : Space n) (v : Space n)]
    ring
  simp_rw [hi, mean, integral_const_mul]
  change mean (fun v : Sphere n => g v * mean (fun u : Sphere n =>
    Real.sign ⟪(v : Space n),(u : Space n)⟫ * ⟪sphereGradient f u,(v : Space n)⟫)) = _
  simp_rw [mean_sign_gradient_radial hf]
  simp only [cosine, mean, ← mul_div_assoc, integral_div, mul_neg]
  rw [integral_neg]
  have hb := (cosineConstant_pos n).ne'
  field_simp

lemma signedPair_tangent_tangent {n : ℕ} [NeZero n] {f g : Space n → ℝ}
    (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 1 g) :
    signedPair (tangentField f) (tangentField g) = ((n : ℝ)-1)*
      signedPair (radialField f) (tangentField g) := by
  rw [signedPair_eq_swap (tangentField_continuous hf) (tangentField_continuous hg),
    signedPair_eq_swap (radialField_continuous hf.continuous) (tangentField_continuous hg)]
  have hi (v : Sphere n) :
      mean (fun u : Sphere n => Real.sign ⟪(u : Space n),(v : Space n)⟫ * ⟪tangentField f u,tangentField g v⟫) =
        ((n : ℝ)-1)*mean (fun u : Sphere n => Real.sign ⟪(u : Space n),(v : Space n)⟫ * ⟪radialField f u,tangentField g v⟫) := by
    have h := mean_sign_gradient_tangent hf (v : Space n) (sphereGradient g v)
      (by rw [real_inner_comm]; exact sphereGradient_tangent g v)
    convert h using 1
    · congr 1
      funext u
      rw [real_inner_comm (u : Space n) (v : Space n)]
      rfl
    · congr 1
      congr 1
      funext u
      simp only [radialField, tangentField, real_inner_smul_left]
      rw [real_inner_comm (u : Space n) (v : Space n)]
      ring
  simp_rw [hi]
  exact integral_const_mul _ _

end PettyProjection.Spherical
end

noncomputable section
open Set MeasureTheory Metric Filter Topology Function
open scoped ENNReal RealInnerProductSpace Gradient
namespace PettyProjection.Spherical

lemma signField_continuous {n : ℕ} (k : ℝ) {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) :
    Continuous (signField k f) :=
  (radialField_continuous hf.continuous).add ((tangentField_continuous hf).const_smul k⁻¹)

lemma signedPair_fields {n : ℕ} (hn : 2 ≤ n) [NeZero n]
    (k : ℝ) {f g : Space n → ℝ} (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) :
    signedPair (signField k f) (signField k g) = cosineConstant n *
      mean (fun u : Sphere n => f u*signOperator (2/k+((n : ℝ)-1)/k^2) g hg u) := by
  have hf1 := hf.of_le (by norm_num : (1 : WithTop ℕ∞) ≤ 2)
  have hg1 := hg.of_le (by norm_num : (1 : WithTop ℕ∞) ≤ 2)
  have hrf := radialField_continuous hf.continuous
  have hrg := radialField_continuous hg.continuous
  have htf := tangentField_continuous hf1
  have htg := tangentField_continuous hg1
  have hktf : Continuous (fun u => k⁻¹ • tangentField f u) := htf.const_smul k⁻¹
  have hktg : Continuous (fun u => k⁻¹ • tangentField g u) := htg.const_smul k⁻¹
  have hrt : signedPair (radialField f) (tangentField g) = -cosineConstant n *
      mean (fun u : Sphere n => f u*cosine (fun v : Sphere n => casimir g v) u) := by
    rw [signedPair_symm hrf htg]
    exact signedPair_tangent_radial hg hf.continuous
  have htr : signedPair (tangentField f) (radialField g) = -cosineConstant n *
      mean (fun u : Sphere n => f u*cosine (fun v : Sphere n => casimir g v) u) := by
    rw [signedPair_tangent_radial hf hg.continuous, cosine_casimir_selfadj hn hg hf]
  change signedPair (fun u => radialField f u+k⁻¹ • tangentField f u)
    (fun u => radialField g u+k⁻¹ • tangentField g u) = _
  rw [signedPair_add_left hrf hktf (show Continuous (fun u => radialField g u+k⁻¹ • tangentField g u) from signField_continuous k hg1),
    signedPair_add_right hrf hrg hktg,
    signedPair_add_right hktf hrg hktg,
    signedPair_smul_right, signedPair_smul_left, signedPair_smul_left, signedPair_smul_right,
    signedPair_radial_radial hf.continuous hg.continuous,
    signedPair_tangent_tangent hf1 hg1, hrt, htr,
    mean_signOperator_expand _ hf.continuous hg]
  simp only [div_eq_mul_inv]
  ring

lemma sign_form_estimate {n : ℕ} (hn : 2 ≤ n) [NeZero n]
    {k : ℝ} (hk : 0 < k) {f g : Space n → ℝ} (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g)
    (h2 : (harmonicSpace n 2).starProjection (toH n (restrictContinuous f hf.continuous)) = 0) :
    mean (fun u : Sphere n => f u)*mean (fun u : Sphere n => g u) -
      ((1+(2/k+((n : ℝ)-1)/k^2)*(4*(n+2)))/((n+1)*(n+3)))*
        ‖Q n (toH n (restrictContinuous f hf.continuous))‖*
        ‖Q n (toH n (restrictContinuous g hg.continuous))‖ ≤
      signedPair (signField k f) (signField k g)/cosineConstant n := by
  rw [signedPair_fields hn k hf hg, mul_div_cancel_left₀ _ (cosineConstant_pos n).ne']
  have hn' : (1 : ℝ) ≤ n := by exact_mod_cast (by omega : 1 ≤ n)
  exact sign_spectral_estimate hn (by positivity) _ hg h2

end PettyProjection.Spherical
end

end OAI