| import OAI.Geometry.Kahler.BaseCauchyJets |
| |
| namespace OAI |
|
|
| universe uKahler8418_1 uKahler8418_2 uKahler8437_1 uKahler8437_2 uKahler8437_3 uKahler8458_1 uKahler8458_2 |
|
|
| open scoped ContDiff |
| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ComplexOrder |
| open Set Filter Topology |
| noncomputable section |
|
|
| open Set Filter Topology |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
| |
| lemma analytic_real_jet_norm {E : Type uKahler8418_1} {F : Type uKahler8418_2} [NormedAddCommGroup E] [NormedSpace ℂ E] |
| [NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F] |
| {f : E → F} {x : E} (hf : AnalyticAt ℂ f x) (n : ℕ) : |
| ‖iteratedFDeriv ℝ n f x‖=‖iteratedFDeriv ℂ n f x‖ := by |
| rw [← (hf.contDiffAt : ContDiffAt ℂ n f x).restrictScalars_iteratedFDeriv (𝕜 := ℝ)] |
| exact ContinuousMultilinearMap.norm_restrictScalars _ |
|
|
| lemma small_power_bound {t B : ℝ} (ht : 0 < t) (ht1 : t ≤ 1) (hB : 1 ≤ B) |
| {i n : ℕ} (hi : 1 ≤ i) (hin : i ≤ n) : |
| B*t ≤ (B*t^((n:ℝ)⁻¹))^i := by |
| have hn : 0 < (n:ℝ) := by exact_mod_cast (show 0 < n by omega) |
| rw [mul_pow,← Real.rpow_mul_natCast ht.le] |
| have hex : (n:ℝ)⁻¹*(i:ℝ) ≤ 1 := by |
| rw [← div_eq_inv_mul] |
| exact (div_le_one hn).mpr (by exact_mod_cast hin) |
| have hp : t ≤ t^((n:ℝ)⁻¹*(i:ℝ)) := by |
| simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_ge ht ht1 hex |
| exact mul_le_mul (by simpa using pow_le_pow_right₀ hB hi) hp ht.le (by positivity) |
|
|
| lemma norm_composition_jet_linear {E : Type uKahler8437_1} {F : Type uKahler8437_2} {G : Type uKahler8437_3} |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] |
| [NormedAddCommGroup G] [NormedSpace ℝ G] |
| {f : E → F} {g : F → G} {s : Set E} (hs : IsOpen s) {x : E} (hx : x ∈ s) |
| {n : ℕ} (hn : 1 ≤ n) (hf : ContDiffOn ℝ n f s) (hg : ContDiff ℝ n g) |
| {B C t : ℝ} (hB : 1 ≤ B) (ht : 0 < t) (ht1 : t ≤ 1) |
| (hC : ∀ i, i ≤ n → ‖iteratedFDeriv ℝ i g (f x)‖ ≤ C) |
| (hbound : ∀ i, 1 ≤ i → i ≤ n → ‖iteratedFDeriv ℝ i f x‖ ≤ B*t) : |
| ‖iteratedFDeriv ℝ n (g ∘ f) x‖ ≤ n.factorial*C*B^n*t := by |
| have hh := norm_iteratedFDerivWithin_comp_le hg.contDiffOn hf le_rfl uniqueDiffOn_univ |
| (hs.uniqueDiffOn) (mapsTo_univ f s) hx |
| (C := C) (D := B*t^((n:ℝ)⁻¹)) |
| (by simpa only [iteratedFDerivWithin_univ] using hC) |
| (fun i hi hin => by |
| rw [iteratedFDerivWithin_of_isOpen i hs hx] |
| exact (hbound i hi hin).trans (small_power_bound ht ht1 hB hi hin)) |
| rw [iteratedFDerivWithin_of_isOpen n hs hx] at hh |
| have hn0 : (n:ℝ) ≠ 0 := by exact_mod_cast (show n ≠ 0 by omega) |
| rw [mul_pow,← Real.rpow_mul_natCast ht.le,inv_mul_cancel₀ hn0,Real.rpow_one] at hh |
| exact hh.trans_eq (by ring) |
|
|
| lemma smooth_jet_bound_compact {E : Type uKahler8458_1} {F : Type uKahler8458_2} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] {g : E → F} (hg : ContDiff ℝ ∞ g) |
| {s : Set E} (hs : IsCompact s) (n : ℕ) : |
| ∃ C : ℝ, 1 ≤ C ∧ ∀ i, i ≤ n → ∀ x ∈ s, ‖iteratedFDeriv ℝ i g x‖ ≤ C := by |
| have hex (i : ℕ) : ∃ C : ℝ, 0 ≤ C ∧ ∀ x ∈ s, ‖iteratedFDeriv ℝ i g x‖ ≤ C := by |
| have hc : Continuous (iteratedFDeriv ℝ i g) := ContDiff.continuous_iteratedFDeriv (by exact_mod_cast (le_top : (i:ℕ∞) ≤ ⊤)) hg |
| obtain ⟨C,hC⟩ := hs.bddAbove_image hc.norm.continuousOn |
| exact ⟨max C 0,le_max_right _ _,fun x hx => (hC (mem_image_of_mem _ hx)).trans (le_max_left _ _)⟩ |
| choose C hC hbound using hex |
| refine ⟨1+∑ j ∈ Finset.range (n+1), C j,?_,?_⟩ |
| · have := Finset.sum_nonneg (s := Finset.range (n+1)) (fun j _ => hC j) |
| linarith |
| · intro i hi x hx |
| have hh := Finset.single_le_sum (s := Finset.range (n+1)) (fun j _ => hC j) (show i ∈ Finset.range (n+1) by simp |
| exact (hbound i x hx).trans (by linarith) |
|
|
| end PinchedHartogs.BaseConstruction |
| |
| end |
|
|
| end OAI |
|
|
