File size: 3,485 Bytes
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 | import OAI.Geometry.IsometricImmersion.Metrics.LocalMetricPCoefficients
namespace OAI
noncomputable section
open Set Filter Function MeasureTheory
open scoped ContDiff Topology BigOperators Matrix ENNReal NNReal
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry SmoothLocal.Flow SmoothLocal.ODE
theorem exists_localized_actualHighRemainder_bound
(G Z M : ℝ) (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hM : 0 ≤ M)
{d c : ℝ} (hd : 0 < d) (hc : 0 < c) (m : ℕ) (hm : 8 ≤ m + 3) :
∃ C : ℝ, 0 ≤ C ∧ ∀ (g : MetricField) (z : Coord → ℝ) (U V : Set Coord),
SmoothPositiveOn g U → IsOpen U → modelSquare ⊆ U →
(∀ i j : Fin 2, ∀ k ≤ 4, ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => g q i j) p‖ ≤ G) →
(∀ p ∈ modelSquare, d ≤ |(g p).det|) →
ContDiffOn ℝ ∞ z U →
(∀ k ≤ 8, ∀ p ∈ modelSquare, ‖iteratedFDeriv ℝ k z p‖ ≤ Z) →
(∀ p ∈ modelSquare, c ≤ |covHessian g z p 1 1|) →
MeasurableSet V → V ⊆ modelSquare → volume V < (⊤ : ℝ≥0∞) →
(∀ i j k, k ≤ m + 5 → ∀ p ∈ V,
‖iteratedFDeriv ℝ k (fun q => g q i j) p‖ ≤ M) →
∀ B : ℝ, 1 ≤ B → ∀ H : ℝ≥0,
(∀ n j, n + heightStateBaseOrder j ≤ m + 1 →
∀ p ∈ V, |heightStateFactor z n j p| ≤ B) →
(∀ n j, n + heightStateBaseOrder j ≤ m + 3 →
eLpNorm (heightStateFactor z n j) 2 (volume.restrict V) ≤ (H : ℝ≥0∞)) →
eLpNorm (actualHighRemainder g z m) 2 (volume.restrict V) ≤
actualHighRemainderL2Budget C (heightPFirstBound G Z d c) B H V m ∧
actualHighRemainderL2Budget C (heightPFirstBound G Z d c) B H V m < (⊤ : ℝ≥0∞) ∧
MemLp (actualHighRemainder g z m) 2 (volume.restrict V) := by
obtain ⟨C, hC, houter⟩ := exists_same_region_remainder_outer_bound M Z hM hd hc m
refine ⟨C, hC, ?_⟩
intro g z U V hg hU hSU hgB hdet hz hzB hyy hV hVS hVfinite hgHigh B hB H hlow hheight
have hgCoord (i j : Fin 2) : CoordinateBound (fun p => g p i j) modelSquare 4 G :=
coordinateBound_of_frechet_bounds (hg.1 i j) hU hSU (hgB i j)
have hzCoord : CoordinateBound z modelSquare 5 Z :=
coordinateBound_five_of_frechet_eight hz hU hSU hzB
obtain ⟨W, hW, hSW, hWU, hne, _, _, _, hfirst, _⟩ :=
exists_open_low_height_coefficients hg hU hSU hz hG hZ hd hc hgCoord hzCoord hdet hyy
have hgW : SmoothPositiveOn g W :=
⟨fun i j => (hg.1 i j).mono hWU, fun p hp => hg.2 p (hWU hp)⟩
have hVW : V ⊆ W := hVS.trans hSW
have hCfirst := heightPFirstBound_nonneg hG hZ hd hc
have ho : ∀ w ∈ topResidualWords m, ∀ r, ∀ p ∈ V,
|coordinateChainCoefficient g z w r p| ≤ C :=
houter g z U V hg hU (hVS.trans hSU) hVS hgHigh
((hzCoord.mono (by norm_num) le_rfl).restrict_domain hVS)
(fun p hp => hdet p (hVS hp)) (fun p hp => hyy p (hVS hp))
have hf : ∀ i : Fin 2, ∀ p ∈ V,
|coordPartial 1 (heightPFirst g z i) p| ≤ heightPFirstBound G Z d c :=
fun i p hp => hfirst i [1] (by norm_num) p (hVS hp)
exact ⟨actualHighRemainder_eLpNorm_two hgW hW (hz.mono hWU) hne hV hVW hm hC hCfirst hB
hlow hheight ho hf,
actualHighRemainderL2Budget_lt_top C (heightPFirstBound G Z d c) B H hVfinite m,
actualHighRemainder_memLp_two hgW hW (hz.mono hWU) hne hV hVW hVfinite hm hC hCfirst hB
hlow hheight ho hf⟩
end SmoothLocal.HighEquation
end
end OAI
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