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59d1c80 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 | import OAI.Geometry.KahlerSplitting.Grid
namespace OAI
section
noncomputable section
open scoped Manifold ContDiff
open Set
namespace UniversalCoverSplitting
open scoped Manifold ContDiff
open Set
def IsBox {n r₁ r₂ : ℕ} {X : ComplexManifold n}
(S : HolomorphicSplitting X r₁ r₂) (U : Set (ℂ × ℂ))
(f : ℂ × ℂ → X.Carrier) : Prop :=
MDifferentiableOn 𝓘(ℂ, ℂ × ℂ) (model n) f U ∧
∀ z ∈ U,
S.projection (f z) (mfderiv 𝓘(ℂ, ℂ × ℂ) (model n) f z (1, 0)) =
mfderiv 𝓘(ℂ, ℂ × ℂ) (model n) f z (1, 0) ∧
S.projection (f z) (mfderiv 𝓘(ℂ, ℂ × ℂ) (model n) f z (0, 1)) = 0
def hartogsFigure (A : (ℂ × ℂ) ≃ᵃ[ℂ] (ℂ × ℂ)) (e a c b : ℝ) : Set (ℂ × ℂ) :=
{z | (‖(A z).1‖ < e ∧ ‖(A z).2‖ < b) ∨
(‖(A z).1‖ < a ∧ c < ‖(A z).2‖ ∧ ‖(A z).2‖ < b)}
def hartogsEnvelope (A : (ℂ × ℂ) ≃ᵃ[ℂ] (ℂ × ℂ)) (a b : ℝ) : Set (ℂ × ℂ) :=
{z | ‖(A z).1‖ < a ∧ ‖(A z).2‖ < b}
def HartogsBoxExtension {n r₁ r₂ : ℕ} {X : ComplexManifold n}
(S : HolomorphicSplitting X r₁ r₂) : Prop :=
∀ (A : (ℂ × ℂ) ≃ᵃ[ℂ] (ℂ × ℂ)) (e a c b : ℝ),
0 < e → e < a → 0 < c → c < b →
∀ f : ℂ × ℂ → X.Carrier, IsBox S (hartogsFigure A e a c b) f →
∃ F : ℂ × ℂ → X.Carrier,
IsBox S (hartogsEnvelope A a b) F ∧
EqOn F f (hartogsFigure A e a c b) ∧
∀ G : ℂ × ℂ → X.Carrier,
IsBox S (hartogsEnvelope A a b) G →
EqOn G f (hartogsFigure A e a c b) → EqOn G F (hartogsEnvelope A a b)
section ProjectionEstimate
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
theorem mixed_direction_identity (P : E →L[ℂ] E) (D : (ℂ × ℂ) →L[ℂ] E)
(h₁ : P (D (1, 0)) = D (1, 0)) (h₂ : P (D (0, 1)) = 0)
(a₁ a₂ b₁ b₂ : ℂ) (hb₁ : b₁ ≠ 0) (hb₂ : b₂ ≠ 0) :
D (a₁, a₂) = (a₁ / b₁) • P (D (b₁, b₂)) +
(a₂ / b₂) • ((ContinuousLinearMap.id ℂ E - P) (D (b₁, b₂))) := by
have split (c₁ c₂ : ℂ) : D (c₁, c₂) = c₁ • D (1, 0) + c₂ • D (0, 1) := by
calc
D (c₁, c₂) = D (c₁ • (1, 0) + c₂ • (0, 1)) := by congr 1; ext <;> simp
_ = c₁ • D (1, 0) + c₂ • D (0, 1) := by simp only [map_add, map_smul]
rw [split a₁ a₂, split b₁ b₂]
simp only [map_add, map_smul, h₁, h₂, smul_zero, add_zero,
sub_apply, ContinuousLinearMap.id_apply]
simp only [sub_self, sub_zero, smul_zero, zero_add, smul_smul,
div_mul_cancel₀ _ hb₁, div_mul_cancel₀ _ hb₂]
theorem mixed_direction_norm_le (P : E →L[ℂ] E) (D : (ℂ × ℂ) →L[ℂ] E)
(h₁ : P (D (1, 0)) = D (1, 0)) (h₂ : P (D (0, 1)) = 0)
(a₁ a₂ b₁ b₂ : ℂ) (hb₁ : b₁ ≠ 0) (hb₂ : b₂ ≠ 0) :
‖D (a₁, a₂)‖ ≤
(‖a₁ / b₁‖ * ‖P‖ + ‖a₂ / b₂‖ * ‖ContinuousLinearMap.id ℂ E - P‖) *
‖D (b₁, b₂)‖ := by
calc
‖D (a₁, a₂)‖ = ‖(a₁ / b₁) • P (D (b₁, b₂)) +
(a₂ / b₂) • ((ContinuousLinearMap.id ℂ E - P) (D (b₁, b₂)))‖ := by
rw [mixed_direction_identity P D h₁ h₂ a₁ a₂ b₁ b₂ hb₁ hb₂]
_ ≤ ‖(a₁ / b₁) • P (D (b₁, b₂))‖ +
‖(a₂ / b₂) • ((ContinuousLinearMap.id ℂ E - P) (D (b₁, b₂)))‖ := norm_add_le _ _
_ = ‖a₁ / b₁‖ * ‖P (D (b₁, b₂))‖ +
‖a₂ / b₂‖ * ‖(ContinuousLinearMap.id ℂ E - P) (D (b₁, b₂))‖ := by
rw [norm_smul, norm_smul]
_ ≤ ‖a₁ / b₁‖ * (‖P‖ * ‖D (b₁, b₂)‖) +
‖a₂ / b₂‖ * (‖ContinuousLinearMap.id ℂ E - P‖ * ‖D (b₁, b₂)‖) := by
exact add_le_add (mul_le_mul_of_nonneg_left (P.le_opNorm _) (norm_nonneg _))
(mul_le_mul_of_nonneg_left ((ContinuousLinearMap.id ℂ E - P).le_opNorm _)
(norm_nonneg _))
_ = _ := by ring
end ProjectionEstimate
end UniversalCoverSplitting
noncomputable section
open Set Topology Filter Metric
namespace UniversalCoverSplitting
section CompactQuadratic
variable {B E : Type*} [TopologicalSpace B] [NormedAddCommGroup E]
[NormedSpace ℝ E] [FiniteDimensional ℝ E]
theorem exists_quadratic_operator_bound {K : Set B} (hK : IsCompact K)
(Q : B → E → ℝ) (P : B → E →L[ℝ] E)
(hQ : ContinuousOn (fun p : B × E ↦ Q p.1 p.2) (K ×ˢ univ))
(hP : ContinuousOn P K)
(hpos : ∀ x ∈ K, ∀ u, u ≠ 0 → 0 < Q x u)
(hhom : ∀ x ∈ K, ∀ (r : ℝ) u, Q x (r • u) = r ^ 2 * Q x u) :
∃ C : ℝ, 0 < C ∧ ∀ x ∈ K, ∀ u, Q x (P x u) ≤ C * Q x u := by
let L : Set (B × E) := K ×ˢ sphere 0 1
have hL : IsCompact L := hK.prod (isCompact_sphere 0 1)
have hQL : ContinuousOn (fun p : B × E ↦ Q p.1 p.2) L :=
hQ.mono (prod_mono_right (subset_univ _))
have hnum : ContinuousOn (fun p : B × E ↦ Q p.1 (P p.1 p.2)) L := by
apply hQ.comp
(continuousOn_fst.prodMk ((hP.comp continuousOn_fst (fun _ h ↦ h.1)).clm_apply
continuousOn_snd))
exact fun p hp ↦ ⟨hp.1, mem_univ _⟩
have hden : ∀ p ∈ L, Q p.1 p.2 ≠ 0 := by
intro p hp
apply ne_of_gt (hpos p.1 hp.1 p.2 _)
intro he
have hn : ‖p.2‖ = (1:ℝ) := by simpa using hp.2
simp [he] at hn
obtain ⟨C,hC⟩ := hL.bddAbove_image (hnum.div hQL hden)
refine ⟨max C 1, by positivity, fun x hx u ↦ ?_⟩
have hzero : Q x 0 = 0 := by simpa using hhom x hx 0 (0:E)
by_cases hu : u = 0
· simp [hu,hzero]
have hn : 0 < ‖u‖ := norm_pos_iff.mpr hu
let v := ‖u‖⁻¹ • u
have hvnorm : ‖v‖ = 1 := by
dsimp [v]
rw [norm_smul_of_nonneg (inv_nonneg.mpr hn.le), inv_mul_cancel₀ hn.ne']
have hv : v ∈ sphere (0:E) 1 := by simpa using hvnorm
have hb : Q x (P x v) / Q x v ≤ max C 1 :=
(hC ⟨(x,v),⟨hx,hv⟩,rfl⟩).trans (le_max_left _ _)
have hvpos : 0 < Q x v := hpos x hx v (by intro hh; simp [hh] at hvnorm)
have hquad : Q x (P x v) ≤ max C 1 * Q x v := (div_le_iff₀ hvpos).mp hb
have huv : ‖u‖ • v = u := by
dsimp [v]
rw [smul_smul, mul_inv_cancel₀ hn.ne', one_smul]
calc
Q x (P x u) = ‖u‖ ^ 2 * Q x (P x v) := by
conv_lhs => rw [← huv, map_smul]
exact hhom x hx ‖u‖ (P x v)
_ ≤ ‖u‖ ^ 2 * (max C 1 * Q x v) := mul_le_mul_of_nonneg_left hquad (sq_nonneg _)
_ = max C 1 * Q x u := by
have hs : Q x u = ‖u‖ ^ 2 * Q x v := by rw [← hhom x hx ‖u‖ v, huv]
rw [hs]
ring
end CompactQuadratic
end UniversalCoverSplitting
end
end
end
end OAI
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