| 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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|
