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import OAI.Geometry.IsometricImmersion.Energy.FiniteSumL2
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped ContDiff Topology Interval BigOperators ENNReal NNReal
namespace SmoothLocal.Hyperbolic
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Analytic
def spatialProduct {ι : Type*} {n : ℕ} (F : ι → Coord → ℝ)
(slot : Fin n → ι) (order : Fin n → ℕ) (p : Coord) : ℝ :=
∏ i, spatialJet (F (slot i)) (order i) p
def spatialSourceTerm {ι : Type*} {n : ℕ} (coeff : Coord → ℝ) (F : ι → Coord → ℝ)
(slot : Fin n → ι) (order : Fin n → ℕ) (p : Coord) : ℝ :=
coeff p * spatialProduct F slot order p
def spatialLowFactorBound (lengthFloor : ℝ) (H : ℝ≥0) : ℝ :=
max 1 (Real.sqrt (2 + 1 / lengthFloor) * (H : ℝ))
def spatialProductL2Budget (lengthFloor left right : ℝ) (H : ℝ≥0) (n : ℕ) : ℝ≥0∞ :=
‖spatialLowFactorBound lengthFloor H ^ n‖ₑ *
((H : ℝ≥0∞) + (volume (Icc left right)) ^ (1 / (2 : ℝ)))
theorem spatialProductL2Budget_lt_top (lengthFloor left right : ℝ) (H : ℝ≥0) (n : ℕ) :
spatialProductL2Budget lengthFloor left right H n < (⊤ : ℝ≥0∞) := by
unfold spatialProductL2Budget
rw [Real.volume_Icc]
finiteness
theorem spatialProduct_contDiffOn
{ι : Type*} {n : ℕ} {F : ι → Coord → ℝ} {U : Set Coord}
(hU : IsOpen U) (hF : ∀ i, ContDiffOn ℝ ∞ (F i) U)
(slot : Fin n → ι) (order : Fin n → ℕ) :
ContDiffOn ℝ ∞ (spatialProduct F slot order) U := by
apply contDiffOn_prod
intro index _
exact spatialJet_contDiffOn hU (hF (slot index)) (order index)
theorem spatialProduct_eLpNorm_two
{ι : Type*} {n N : ℕ} {F : ι → Coord → ℝ} {U : Set Coord}
{theta left right lengthFloor : ℝ} {H : ℝ≥0}
(hU : IsOpen U) (hF : ∀ i, ContDiffOn ℝ ∞ (F i) U)
(hseg : ∀ x ∈ Icc left right, coordinatePoint x theta ∈ U)
(hlength : 0 < lengthFloor) (hwidth : lengthFloor ≤ right - left)
(hL2 : SpatialSliceL2Bound F theta left right N H)
(slot : Fin n → ι) (order : Fin n → ℕ)
(horder : ∀ i, order i ≤ N)
(hunique : ∀ i j, order i = N → order j = N → i = j) :
eLpNorm (fun x => spatialProduct F slot order (coordinatePoint x theta)) 2
(volume.restrict (Icc left right)) ≤ spatialProductL2Budget lengthFloor left right H n := by
classical
have hmeas := (smooth_slice_memLp_two
(spatialProduct_contDiffOn hU hF slot order).continuousOn hseg).aestronglyMeasurable
let B := spatialLowFactorBound lengthFloor H
have hB1 : 1 ≤ B := le_max_left _ _
have hB0 : 0 ≤ B := zero_le_one.trans hB1
have hlow (i : Fin n) (hi : order i + 1 ≤ N) (x : ℝ) (hx : x ∈ Icc left right) :
‖spatialJet (F (slot i)) (order i) (coordinatePoint x theta)‖ ≤ B := by
rw [Real.norm_eq_abs]
exact (hL2.pointwise_below_top hU hF hseg hlength hwidth (slot i) (order i) hi hx).trans
(le_max_right _ _)
by_cases hh : ∃ j, order j = N
· obtain ⟨j, hj⟩ := hh
have hb (i : Fin n) (hi : i ∈ (Finset.univ : Finset (Fin n)).erase j)
(x : ℝ) (hx : x ∈ Icc left right) :
‖spatialJet (F (slot i)) (order i) (coordinatePoint x theta)‖ ≤ B := by
apply hlow i _ x hx
have hi_ne : i ≠ j := (Finset.mem_erase.mp hi).1
have hnot : order i ≠ N := fun hie => hi_ne (hunique i j hie hj)
have hoi := horder i
omega
have hprod := finite_product_eLpNorm_two_of_aestronglyMeasurable (μ := volume) Finset.univ
(fun i x => spatialJet (F (slot i)) (order i) (coordinatePoint x theta))
(Finset.mem_univ j) hB0 measurableSet_Icc hb hmeas
have hcard : ((Finset.univ : Finset (Fin n)).erase j).card ≤ n := by
simpa only [Finset.card_univ, Fintype.card_fin] using
Finset.card_le_card (Finset.erase_subset j (Finset.univ : Finset (Fin n)))
have hpow : ‖B ^ ((Finset.univ : Finset (Fin n)).erase j).card‖ₑ ≤ ‖B ^ n‖ₑ := by
rw [enorm_le_iff_norm_le, Real.norm_of_nonneg (pow_nonneg hB0 _),
Real.norm_of_nonneg (pow_nonneg hB0 _)]
exact pow_le_pow_right₀ hB1 hcard
calc
_ ≤ ‖B ^ ((Finset.univ : Finset (Fin n)).erase j).card‖ₑ *
eLpNorm (fun x => spatialJet (F (slot j)) (order j) (coordinatePoint x theta)) 2
(volume.restrict (Icc left right)) := hprod
_ ≤ ‖B ^ n‖ₑ * (H : ℝ≥0∞) := mul_le_mul' hpow (hL2 (slot j) (order j) (horder j))
_ ≤ _ := mul_le_mul' le_rfl (le_add_of_nonneg_right (by positivity))
· have hb (i : Fin n) (_ : i ∈ (Finset.univ : Finset (Fin n)))
(x : ℝ) (hx : x ∈ Icc left right) :
‖spatialJet (F (slot i)) (order i) (coordinatePoint x theta)‖ ≤ B := by
apply hlow i _ x hx
have hnot : order i ≠ N := fun hie => hh ⟨i, hie⟩
have hoi := horder i
omega
have hprod := finite_product_bounded_eLpNorm_two_of_aestronglyMeasurable
(μ := volume) Finset.univ
(fun i x => spatialJet (F (slot i)) (order i) (coordinatePoint x theta))
hB0 measurableSet_Icc hb hmeas
calc
_ ≤ ‖B ^ n‖ₑ * (volume (Icc left right)) ^ (1 / (2 : ℝ)) := by
simpa only [spatialProduct, Finset.card_univ, Fintype.card_fin] using hprod
_ ≤ _ := mul_le_mul' le_rfl (le_add_of_nonneg_left (by positivity))
theorem real_interval_coefficient_eLpNorm_two
{a f : ℝ → ℝ} {left right C : ℝ} (hC : 0 ≤ C)
(ha : ∀ x ∈ Icc left right, |a x| ≤ C) :
eLpNorm' (fun x => a x * f x) (2 : ℝ) (volume.restrict (Icc left right)) ≤
‖C‖ₑ * eLpNorm' f (2 : ℝ) (volume.restrict (Icc left right)) := by
have hdom : ∀ᵐ x ∂volume.restrict (Icc left right), ‖a x * f x‖ ≤ ‖C * f x‖ := by
filter_upwards [ae_restrict_mem measurableSet_Icc] with x hx
simp only [norm_mul, Real.norm_eq_abs, abs_of_nonneg hC]
exact mul_le_mul_of_nonneg_right (ha x hx) (abs_nonneg _)
calc
_ ≤ eLpNorm' (fun x => C * f x) (2 : ℝ) (volume.restrict (Icc left right)) :=
eLpNorm'_mono_ae (by norm_num) hdom
_ = _ := eLpNorm'_const_smul (𝕜 := ℝ) (F := ℝ) (f := f) C (by norm_num)
theorem real_interval_coefficient_eLpNorm_two_of_aestronglyMeasurable
{coefficient factor : ℝ → ℝ} {left right bound : ℝ} (hbound : 0 ≤ bound)
(hcoefficient : ∀ point ∈ Icc left right, |coefficient point| ≤ bound)
(hmeas : AEStronglyMeasurable (fun point => coefficient point * factor point)
(volume.restrict (Icc left right))) :
eLpNorm (fun point => coefficient point * factor point) 2
(volume.restrict (Icc left right)) ≤
‖bound‖ₑ * eLpNorm factor 2 (volume.restrict (Icc left right)) := by
have hfactor : eLpNorm' factor (2 : ℝ) (volume.restrict (Icc left right)) ≤
eLpNorm factor 2 (volume.restrict (Icc left right)) := by
by_cases hfactorMeas : AEStronglyMeasurable factor (volume.restrict (Icc left right))
· rw [eLpNorm_eq_eLpNorm' (by norm_num) (by norm_num) hfactorMeas]
norm_num
· rw [eLpNorm_of_not_aestronglyMeasurable hfactorMeas]
exact le_top
rw [eLpNorm_eq_eLpNorm' (by norm_num) (by norm_num) hmeas, ENNReal.toReal_ofNat]
exact (real_interval_coefficient_eLpNorm_two hbound hcoefficient).trans
(mul_le_mul_right hfactor _)
theorem spatialSourceTerm_memLp_two
{ι : Type*} {n : ℕ} {F : ι → Coord → ℝ} {coeff : Coord → ℝ} {U : Set Coord}
{theta left right : ℝ}
(hU : IsOpen U) (hF : ∀ i, ContDiffOn ℝ ∞ (F i) U)
(hc : ContDiffOn ℝ ∞ coeff U)
(hseg : ∀ x ∈ Icc left right, coordinatePoint x theta ∈ U)
(slot : Fin n → ι) (order : Fin n → ℕ) :
MemLp (fun x => spatialSourceTerm coeff F slot order (coordinatePoint x theta)) 2
(volume.restrict (Icc left right)) :=
smooth_slice_memLp_two (hc.mul (spatialProduct_contDiffOn hU hF slot order)).continuousOn hseg
theorem spatialSourceTerm_eLpNorm_two
{ι : Type*} {n N : ℕ} {F : ι → Coord → ℝ} {coeff : Coord → ℝ} {U : Set Coord}
{theta left right lengthFloor C : ℝ} {H : ℝ≥0}
(hU : IsOpen U) (hF : ∀ i, ContDiffOn ℝ ∞ (F i) U)
(hseg : ∀ x ∈ Icc left right, coordinatePoint x theta ∈ U)
(hlength : 0 < lengthFloor) (hwidth : lengthFloor ≤ right - left)
(hL2 : SpatialSliceL2Bound F theta left right N H) (hC : 0 ≤ C)
(hc : ∀ x ∈ Icc left right, |coeff (coordinatePoint x theta)| ≤ C)
(slot : Fin n → ι) (order : Fin n → ℕ)
(horder : ∀ i, order i ≤ N)
(hunique : ∀ i j, order i = N → order j = N → i = j) :
eLpNorm' (fun x => spatialSourceTerm coeff F slot order (coordinatePoint x theta)) (2 : ℝ)
(volume.restrict (Icc left right)) ≤
‖C‖ₑ * spatialProductL2Budget lengthFloor left right H n := by
have hproduct := spatialProduct_eLpNorm_two hU hF hseg hlength hwidth hL2
slot order horder hunique
have hmeas := (smooth_slice_memLp_two
(spatialProduct_contDiffOn hU hF slot order).continuousOn hseg).aestronglyMeasurable
rw [eLpNorm_eq_eLpNorm' (by norm_num) (by norm_num) hmeas,
ENNReal.toReal_ofNat] at hproduct
exact (real_interval_coefficient_eLpNorm_two hC hc).trans
(mul_le_mul' le_rfl hproduct)
theorem spatialSourceTerm_eLpNorm_two_of_aestronglyMeasurable
{ι : Type*} {n N : ℕ} {F : ι → Coord → ℝ} {coeff : Coord → ℝ} {U : Set Coord}
{theta left right lengthFloor C : ℝ} {H : ℝ≥0}
(hU : IsOpen U) (hF : ∀ i, ContDiffOn ℝ ∞ (F i) U)
(hseg : ∀ x ∈ Icc left right, coordinatePoint x theta ∈ U)
(hlength : 0 < lengthFloor) (hwidth : lengthFloor ≤ right - left)
(hL2 : SpatialSliceL2Bound F theta left right N H) (hC : 0 ≤ C)
(hc : ∀ x ∈ Icc left right, |coeff (coordinatePoint x theta)| ≤ C)
(slot : Fin n → ι) (order : Fin n → ℕ)
(horder : ∀ i, order i ≤ N)
(hunique : ∀ i j, order i = N → order j = N → i = j)
(hmeas : AEStronglyMeasurable
(fun x => spatialSourceTerm coeff F slot order (coordinatePoint x theta))
(volume.restrict (Icc left right))) :
eLpNorm (fun x => spatialSourceTerm coeff F slot order (coordinatePoint x theta)) 2
(volume.restrict (Icc left right)) ≤
‖C‖ₑ * spatialProductL2Budget lengthFloor left right H n := by
rw [eLpNorm_eq_eLpNorm' (by norm_num) (by norm_num) hmeas, ENNReal.toReal_ofNat]
exact spatialSourceTerm_eLpNorm_two hU hF hseg hlength hwidth hL2 hC hc
slot order horder hunique
def spatialSourceSum {ι : Type*} {m : ℕ} (arity : Fin m → ℕ)
(coeff : Fin m → Coord → ℝ) (F : ι → Coord → ℝ)
(slot : (k : Fin m) → Fin (arity k) → ι)
(order : (k : Fin m) → Fin (arity k) → ℕ) (p : Coord) : ℝ :=
∑ k, spatialSourceTerm (coeff k) F (slot k) (order k) p
theorem spatialSourceSum_eLpNorm_two
{ι : Type*} {m N : ℕ} {F : ι → Coord → ℝ} {coeff : Fin m → Coord → ℝ} {U : Set Coord}
{theta left right lengthFloor : ℝ} {H : ℝ≥0}
(hU : IsOpen U) (hF : ∀ i, ContDiffOn ℝ ∞ (F i) U)
(hcSmooth : ∀ k, ContDiffOn ℝ ∞ (coeff k) U)
(hseg : ∀ x ∈ Icc left right, coordinatePoint x theta ∈ U)
(hlength : 0 < lengthFloor) (hwidth : lengthFloor ≤ right - left)
(hL2 : SpatialSliceL2Bound F theta left right N H) (C : Fin m → ℝ)
(hC : ∀ k, 0 ≤ C k)
(hc : ∀ k x, x ∈ Icc left right → |coeff k (coordinatePoint x theta)| ≤ C k)
(arity : Fin m → ℕ) (slot : (k : Fin m) → Fin (arity k) → ι)
(order : (k : Fin m) → Fin (arity k) → ℕ)
(horder : ∀ k i, order k i ≤ N)
(hunique : ∀ k i j, order k i = N → order k j = N → i = j) :
eLpNorm (fun x => spatialSourceSum arity coeff F slot order (coordinatePoint x theta)) 2
(volume.restrict (Icc left right)) ≤
∑ k, ‖C k‖ₑ * spatialProductL2Budget lengthFloor left right H (arity k) := by
exact finite_sum_eLpNorm_two (μ := volume) (U := Icc left right) Finset.univ
(fun k x => spatialSourceTerm (coeff k) F (slot k) (order k) (coordinatePoint x theta))
(fun k => ‖C k‖ₑ * spatialProductL2Budget lengthFloor left right H (arity k))
(fun k _ => (spatialSourceTerm_memLp_two hU hF (hcSmooth k) hseg
(slot k) (order k)).aestronglyMeasurable)
(fun k _ => spatialSourceTerm_eLpNorm_two_of_aestronglyMeasurable
hU hF hseg hlength hwidth hL2 (hC k)
(hc k) (slot k) (order k) (horder k) (hunique k)
(spatialSourceTerm_memLp_two hU hF (hcSmooth k) hseg
(slot k) (order k)).aestronglyMeasurable)
end SmoothLocal.Hyperbolic
end
end OAI
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