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110f4f3 | 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 | import Mathlib
import OAI.Analysis.BiholderTransport.Regularity.UniformFiniteGrowth
import OAI.Analysis.BiholderTransport.Calculus.PartialDerivatives
import OAI.Analysis.BiholderTransport.Calculus.TaylorBound
namespace OAI
noncomputable section
namespace WeakMTWTransport
section
open Set Filter
open scoped Topology ContDiff
variable {P E : Type*} [TopologicalSpace P] [CompactSpace P]
[NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
{ι : Type*} [Fintype ι] [Nonempty ι]
lemma compact_finite_smooth_transverse_growth
(f : P → ι → E → ℝ) (w : P → ι → ℝ)
(hw : ∀ a i, 0 ≤ w a i) (hsum : ∀ a, ∑ i, w a i = 1)
(hwc : ∀ i, Continuous (fun a => w a i))
(hf : ∀ a i, ∀ᶠ q : E × P in 𝓝 (0,a), ContDiffAt ℝ 2 (f q.2 i) q.1)
(hf0 : ∀ a i, f a i 0 = 0)
(hLc : ∀ i, Continuous (fun a => fderiv ℝ (f a i) 0))
(hBc : ∀ i, Continuous (fun a => fderiv ℝ (fderiv ℝ (f a i)) 0))
(hD : ∀ a i, ContinuousAt (fun q : E × P =>
fderiv ℝ (fderiv ℝ (f q.2 i)) q.1) (0,a))
(hmean : ∀ a v, ∑ i, w a i * fderiv ℝ (f a i) 0 v = 0)
(hpos : ∀ a v, ‖v‖ = 1 → (∀ i, fderiv ℝ (f a i) 0 v ≤ 0) →
0 < ∑ i, w a i * fderiv ℝ (fderiv ℝ (f a i)) 0 v v) :
∃ b > 0, ∀ᶠ h : E in 𝓝 0, ∀ a : P, ∃ i, b * ‖h‖^2 ≤ f a i h := by
let L := fun a i => fderiv ℝ (f a i) 0
let B := fun a i => (1/2 : ℝ) • fderiv ℝ (fderiv ℝ (f a i)) 0
have hB : ∀ i, Continuous (fun a => B a i) :=
fun i => (continuous_const : Continuous (fun _ : P => (1/2 : ℝ))).smul (hBc i)
have hpos' : ∀ a v, ‖v‖ = 1 → (∀ i, L a i v ≤ 0) →
0 < ∑ i, w a i * B a i v v := by
intro a v hv hl
have he : (∑ i, w a i * B a i v v) = (1/2:ℝ) *
∑ i, w a i * fderiv ℝ (fderiv ℝ (f a i)) 0 v v := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
simp only [B,_root_.smul_apply,smul_eq_mul]
ring
rw [he]
exact mul_pos (by norm_num) (hpos a v hv hl)
obtain ⟨b,hb,H⟩ := uniform_finite_transverse_quadratic_growth w L B hw hsum hmean
hwc hLc hB hpos'
have HT : ∀ᶠ h : E in 𝓝 0, ∀ i, ∀ a : P,
|f a i h - L a i h - B a i h h| ≤ (b/2) * ‖h‖^2 := by
apply Filter.eventually_all.mpr
intro i
have hi := compact_parametric_second_order_remainder (fun a => f a i)
(fun a => hf a i) (fun a => hD a i) (half_pos hb)
filter_upwards [hi] with h hh
intro a
simpa only [hf0,sub_zero,L,B,_root_.smul_apply,smul_eq_mul,one_div,div_eq_mul_inv,
one_mul,mul_comm] using hh a
refine ⟨b/2,half_pos hb,?_⟩
filter_upwards [H,HT] with h hh ht
intro a
obtain ⟨i,hi⟩ := hh a
have he := (abs_le.mp (ht i a)).1
exact ⟨i,by nlinarith⟩
end
open Set Filter
open scoped Topology ContDiff
lemma compact_smooth_data_transverse_growth
{P Q E : Type*} [TopologicalSpace P] [CompactSpace P]
[NormedAddCommGroup Q] [NormedSpace ℝ Q]
[NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
{ι : Type*} [Fintype ι] [Nonempty ι]
(F : Q → ι → E → ℝ) (d : P → Q) (hd : Continuous d) (w : P → ι → ℝ)
(hw : ∀ a i, 0 ≤ w a i) (hsum : ∀ a, ∑ i, w a i = 1)
(hwc : ∀ i, Continuous (fun a => w a i))
(hf : ∀ a i, ContDiffAt ℝ ∞ (fun q : Q × E => F q.1 i q.2) (d a,0))
(hf0 : ∀ a i, F (d a) i 0 = 0)
(hmean : ∀ a v, ∑ i, w a i * fderiv ℝ (F (d a) i) 0 v = 0)
(hpos : ∀ a v, ‖v‖ = 1 → (∀ i, fderiv ℝ (F (d a) i) 0 v ≤ 0) →
0 < ∑ i, w a i * fderiv ℝ (fderiv ℝ (F (d a) i)) 0 v v) :
∃ b > 0, ∀ᶠ h : E in 𝓝 0, ∀ a : P, ∃ i, b * ‖h‖^2 ≤ F (d a) i h := by
have hc : Continuous (fun q : E × P => (d q.2,q.1)) :=
(hd.comp continuous_snd).prodMk continuous_fst
have hF : ∀ a i, ∀ᶠ q : E × P in 𝓝 (0,a), ContDiffAt ℝ 2 (F (d q.2) i) q.1 := by
intro a i
have H := ((hf a i).of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)).eventually (by simp)
filter_upwards [hc.continuousAt.tendsto.eventually H] with q hq
exact hq.comp q.1 (contDiffAt_const.prodMk contDiffAt_id)
have hD : ∀ a i, ContinuousAt (fun q : E × P =>
fderiv ℝ (fderiv ℝ (F (d q.2) i)) q.1) (0,a) := by
intro a i
exact (ContDiffAt.partial_snd_fderiv_two (f := fun q => F q i) (hf a i)).continuousAt.comp
(x := (0,a)) (f := fun q : E × P => (d q.2,q.1)) hc.continuousAt
apply compact_finite_smooth_transverse_growth (fun a => F (d a)) w hw hsum hwc hF hf0
· intro i
apply continuous_iff_continuousAt.mpr
intro a
exact (ContDiffAt.partial_snd_fderiv (f := fun q => F q i) (hf a i)).continuousAt.comp
(x := a) (f := fun a => (d a,(0:E))) (hd.continuousAt.prodMk continuousAt_const)
· intro i
apply continuous_iff_continuousAt.mpr
intro a
exact (hD a i).comp (x := a) (f := fun a => ((0:E),a))
(continuousAt_const.prodMk continuousAt_id)
· exact hD
· exact hmean
· exact hpos
end WeakMTWTransport
end
end OAI
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