| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphDegrees | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphJetMul | |
| import OAI.MeasureTheory.FalconerCompletion.Sphere.ScalarCapPartition | |
| namespace OAI | |
| /-! Graph energies and pair estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace CurrentFalconer.TreeND | |
| open Set MeasureTheory | |
| variable {d : ℕ} | |
| noncomputable def graphDescendants (a p : ℕ) (X : CellPath d) : Finset (CellPath d) := by | |
| classical | |
| exact (dyadicPaths p).filter fun P => P.take a=X | |
| noncomputable def graphChildren (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (a p : ℕ) (X : CellPath d) : Finset (CellPath d) := by | |
| classical | |
| exact (graphDescendants a p X).filter fun P => 0 < ρ (treeCell P) | |
| lemma graphDescendants_subset {a p : ℕ} {X P : CellPath d} (hP : P ∈ graphDescendants a p X) : | |
| treeCell P ⊆ treeCell X := by | |
| classical | |
| rcases Finset.mem_filter.mp hP with ⟨hP,he⟩ | |
| simpa only [he] using treeCell_subset_take P a | |
| lemma graphDescendants_cover {a p : ℕ} (hap : a ≤ p) {X : CellPath d} | |
| (hX : X ∈ dyadicPaths a) : | |
| (⋃ P ∈ graphDescendants a p X, treeCell P)=treeCell X := by | |
| classical | |
| apply Set.Subset.antisymm | |
| · intro x hx | |
| obtain ⟨P,hP,hx⟩ := Set.mem_iUnion₂.mp hx | |
| exact graphDescendants_subset hP hx | |
| · intro x hx | |
| have hxroot : x ∈ treeCell ([] : CellPath d) := by | |
| simpa only [List.take_zero] using treeCell_subset_take X 0 hx | |
| rw [←treeCell_cover_root p] at hxroot | |
| obtain ⟨P,hP,hxP⟩ := Set.mem_iUnion₂.mp hxroot | |
| have hlP : P.length=p := (mem_dyadicPaths _ _).mp hP | |
| have hlX : X.length=a := (mem_dyadicPaths _ _).mp hX | |
| have he : P.take a=X := by | |
| by_contra hne | |
| have hh := treeCell_disjoint (p:=P.take a) (q:=X) | |
| (by simp [List.length_take,hlP,hlX,Nat.min_eq_left hap]) hne | |
| exact Set.disjoint_left.mp hh (treeCell_subset_take P a hxP) hx | |
| exact Set.mem_iUnion₂.mpr ⟨P,Finset.mem_filter.mpr ⟨hP,he⟩,hxP⟩ | |
| lemma graphDescendants_disjoint {a p : ℕ} {X P Q : CellPath d} | |
| (hP : P ∈ graphDescendants a p X) (hQ : Q ∈ graphDescendants a p X) (hne : P ≠ Q) : | |
| Disjoint (treeCell P) (treeCell Q) := by | |
| classical | |
| apply treeCell_disjoint _ hne | |
| exact ((mem_dyadicPaths _ _).mp (Finset.mem_filter.mp hP).1).trans | |
| ((mem_dyadicPaths _ _).mp (Finset.mem_filter.mp hQ).1).symm | |
| lemma graphChildren_integral (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| {a p : ℕ} (hap : a ≤ p) {X : CellPath d} (hX : X ∈ dyadicPaths a) | |
| (F : EuclideanSpace ℝ (Fin d) → ℂ) (hF : IntegrableOn F (treeCell X) ρ) : | |
| (∫ x in treeCell X, F x ∂ρ) = | |
| ∑ P ∈ graphChildren ρ a p X, ∫ x in treeCell P, F x ∂ρ := by | |
| classical | |
| rw [←graphDescendants_cover hap hX] | |
| rw [integral_biUnion_finset _ (fun P _ => measurableSet_treeCell P) | |
| (fun P hP Q hQ hne => graphDescendants_disjoint hP hQ hne) | |
| (fun P hP => hF.mono_set (graphDescendants_subset hP))] | |
| symm | |
| apply Finset.sum_subset (Finset.filter_subset _ _) | |
| intro P hP hnP | |
| have hz : ρ (treeCell P)=0 := by | |
| apply le_antisymm _ bot_le | |
| by_contra hp | |
| exact hnP (Finset.mem_filter.mpr ⟨hP,lt_of_not_ge hp⟩) | |
| rw [Measure.restrict_eq_zero.mpr hz,integral_zero_measure] | |
| lemma graphChildren_mass (ρ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] | |
| {a p : ℕ} (hap : a ≤ p) {X : CellPath d} (hX : X ∈ dyadicPaths a) : | |
| ∑ P ∈ graphChildren ρ a p X, ρ.real (treeCell P) = ρ.real (treeCell X) := by | |
| classical | |
| have h := graphChildren_integral ρ hap hX (fun _ => (1:ℂ)) (integrableOn_const (C:= (1:ℂ))) | |
| have hh := congrArg Complex.re h | |
| simpa [integral_const,Measure.real,Complex.re_sum] using hh.symm | |
| lemma graphChildren_nonempty (ρ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] | |
| {a p : ℕ} (hap : a ≤ p) {X : CellPath d} (hX : X ∈ dyadicPaths a) | |
| (hocc : 0 < ρ (treeCell X)) : (graphChildren ρ a p X).Nonempty := by | |
| have hm := graphChildren_mass ρ hap hX | |
| by_contra he | |
| rw [Finset.not_nonempty_iff_eq_empty.mp he,Finset.sum_empty] at hm | |
| have hp : 0 < ρ.real (treeCell X) := ENNReal.toReal_pos hocc.ne' (measure_ne_top _ _) | |
| linarith | |
| lemma graphChildren_card_le (ρ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] | |
| {a p : ℕ} (hap : a ≤ p) {X : CellPath d} (hX : X ∈ dyadicPaths a) | |
| {l u : ℝ} (hl : 0 < l) | |
| (hlo : ∀ P ∈ graphChildren ρ a p X, l ≤ ρ.real (treeCell P)) | |
| (hup : ρ.real (treeCell X) ≤ u) : | |
| ((graphChildren ρ a p X).card : ℝ) ≤ u/l := by | |
| rw [le_div_iff₀ hl] | |
| calc | |
| _ = ∑ P ∈ graphChildren ρ a p X, l := by simp | |
| _ ≤ ∑ P ∈ graphChildren ρ a p X, ρ.real (treeCell P) := Finset.sum_le_sum hlo | |
| _ = ρ.real (treeCell X) := graphChildren_mass ρ hap hX | |
| _ ≤ u := hup | |
| end CurrentFalconer.TreeND | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace CurrentFalconer.TreeND | |
| open Set MeasureTheory | |
| open scoped RealInnerProductSpace | |
| variable {d : ℕ} | |
| lemma treeCell_diameter {P : CellPath d} {x y : EuclideanSpace ℝ (Fin d)} | |
| (hx : x ∈ treeCell P) (hy : y ∈ treeCell P) : | |
| ‖x-y‖ ≤ (d+1:ℝ)*(2:ℝ)^(-(P.length:ℝ)) := by | |
| have h := ambientGridCell_dist_le hy hx | |
| rw [Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2), Real.rpow_natCast] | |
| simpa only [dist_eq_norm,div_eq_mul_inv] using h | |
| lemma graphChildren_length {ρ : Measure (EuclideanSpace ℝ (Fin d))} {a p : ℕ} {X P : CellPath d} | |
| (hP : P ∈ graphChildren ρ a p X) : P.length=p := by | |
| classical | |
| exact (mem_dyadicPaths _ _).mp (Finset.mem_filter.mp (Finset.mem_filter.mp hP).1).1 | |
| lemma graphChildren_subset {ρ : Measure (EuclideanSpace ℝ (Fin d))} {a p : ℕ} {X P : CellPath d} | |
| (hP : P ∈ graphChildren ρ a p X) : treeCell P ⊆ treeCell X := by | |
| classical | |
| exact graphDescendants_subset (Finset.mem_filter.mp hP).1 | |
| lemma graphChildren_center_error {ρ κ : Measure (EuclideanSpace ℝ (Fin d))} {a p : ℕ} | |
| {X Y P Q : CellPath d} (hP : P ∈ graphChildren ρ a p X) (hQ : Q ∈ graphChildren κ a p Y) | |
| {x y : EuclideanSpace ℝ (Fin d)} (hx : x ∈ treeCell P) (hy : y ∈ treeCell Q) : | |
| ‖(x-y)-(treeCenter P-treeCenter Q)‖ ≤ 2*(d+1:ℝ)*(2:ℝ)^(-(p:ℝ)) := by | |
| have hx' := treeCell_diameter hx (treeCenter_mem P) | |
| have hy' := treeCell_diameter hy (treeCenter_mem Q) | |
| rw [graphChildren_length hP] at hx' | |
| rw [graphChildren_length hQ] at hy' | |
| calc | |
| _ = ‖(x-treeCenter P)-(y-treeCenter Q)‖ := by congr 1; module | |
| _ ≤ ‖x-treeCenter P‖+‖y-treeCenter Q‖ := norm_sub_le _ _ | |
| _ ≤ _ := by linarith | |
| lemma graphChildren_direct_size {ρ κ : Measure (EuclideanSpace ℝ (Fin d))} {a p : ℕ} | |
| {X P Q : CellPath d} (hX : X ∈ dyadicPaths a) | |
| (hP : P ∈ graphChildren ρ a p X) (hQ : Q ∈ graphChildren κ a p X) | |
| {x y : EuclideanSpace ℝ (Fin d)} (hx : x ∈ treeCell P) (hy : y ∈ treeCell Q) : | |
| ‖x-y‖ ≤ (d+1:ℝ)*(2:ℝ)^(-(a:ℝ)) := by | |
| simpa only [(mem_dyadicPaths _ _).mp hX] using | |
| treeCell_diameter (graphChildren_subset hP hx) (graphChildren_subset hQ hy) | |
| lemma graphChildren_cross_size {ρ κ : Measure (EuclideanSpace ℝ (Fin d))} {a p : ℕ} | |
| {X Y P Q : CellPath d} (hX : X ∈ dyadicPaths a) (hY : Y ∈ dyadicPaths a) | |
| (hP : P ∈ graphChildren ρ a p X) (hQ : Q ∈ graphChildren κ a p Y) | |
| {G : ℝ} (hcenters : ‖treeCenter X-treeCenter Y‖ ≤ G*(2:ℝ)^(-(a:ℝ))) | |
| {x y : EuclideanSpace ℝ (Fin d)} (hx : x ∈ treeCell P) (hy : y ∈ treeCell Q) : | |
| ‖x-y‖ ≤ (2*(d+1:ℝ)+G)*(2:ℝ)^(-(a:ℝ)) := by | |
| have hx' := treeCell_diameter (graphChildren_subset hP hx) (treeCenter_mem X) | |
| have hy' := treeCell_diameter (treeCenter_mem Y) (graphChildren_subset hQ hy) | |
| rw [(mem_dyadicPaths _ _).mp hX] at hx' | |
| rw [(mem_dyadicPaths _ _).mp hY] at hy' | |
| calc | |
| _ = ‖(x-treeCenter X)+(treeCenter X-treeCenter Y)+(treeCenter Y-y)‖ := by congr 1; module | |
| _ ≤ ‖x-treeCenter X‖+‖treeCenter X-treeCenter Y‖+‖treeCenter Y-y‖ := norm_add₃_le | |
| _ ≤ _ := by linarith | |
| lemma graphChildren_profile_card (ρ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] | |
| {a p : ℕ} (hap : a ≤ p) {X : CellPath d} (hX : X ∈ dyadicPaths a) | |
| {R ma mp e : ℝ} (hR : 0 < R) | |
| (hlo : ∀ P ∈ graphChildren ρ a p X, R^(-mp-e) ≤ ρ.real (treeCell P)) | |
| (hup : ρ.real (treeCell X) ≤ R^(-ma+e)) : | |
| ((graphChildren ρ a p X).card : ℝ) ≤ R^(mp-ma+2*e) := by | |
| have h := graphChildren_card_le ρ hap hX (Real.rpow_pos_of_pos hR _) hlo hup | |
| rw [←Real.rpow_sub hR] at h | |
| convert h using 1; ring_nf | |
| lemma graphChildren_pair_profile_card | |
| (ρ κ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ] | |
| {a p : ℕ} (hap : a ≤ p) {X Y : CellPath d} (hX : X ∈ dyadicPaths a) (hY : Y ∈ dyadicPaths a) | |
| {R ma₁ mp₁ ma₂ mp₂ e : ℝ} (hR : 0 < R) | |
| (hlo₁ : ∀ P ∈ graphChildren ρ a p X, R^(-mp₁-e) ≤ ρ.real (treeCell P)) | |
| (hlo₂ : ∀ P ∈ graphChildren κ a p Y, R^(-mp₂-e) ≤ κ.real (treeCell P)) | |
| (hup₁ : ρ.real (treeCell X) ≤ R^(-ma₁+e)) | |
| (hup₂ : κ.real (treeCell Y) ≤ R^(-ma₂+e)) : | |
| (graphChildren ρ a p X).card * ((graphChildren κ a p Y).card:ℝ) ≤ | |
| R^((mp₁-ma₁)+(mp₂-ma₂)+4*e) := by | |
| have h₁ := graphChildren_profile_card ρ hap hX hR hlo₁ hup₁ | |
| have h₂ := graphChildren_profile_card κ hap hY hR hlo₂ hup₂ | |
| calc | |
| _ ≤ R^(mp₁-ma₁+2*e)*R^(mp₂-ma₂+2*e) := mul_le_mul h₁ h₂ (by positivity) (by positivity) | |
| _ = _ := by rw [←Real.rpow_add hR]; congr 1; ring | |
| end CurrentFalconer.TreeND | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace CurrentFalconer | |
| open Set MeasureTheory | |
| variable {d : ℕ} | |
| lemma tensorCapND_nonneg (δ : ℝ) (a : Fin d → ℤ) (w : EuclideanSpace ℝ (Fin d)) : | |
| 0 ≤ tensorCapND δ a w := tensorCapKernelND_nonneg _ _ | |
| lemma tensorCapND_le_one (δ : ℝ) (a : Fin d → ℤ) (w : EuclideanSpace ℝ (Fin d)) : | |
| tensorCapND δ a w ≤ 1 := tensorCapKernelND_le_one _ _ | |
| lemma sphereCapGridND_card_bound {δ : ℝ} (hδ : 0 < δ) (hδ1 : δ ≤ 1) : | |
| ((sphereCapGridND d δ).card : ℝ) ≤ (9*δ⁻¹)^d := by | |
| classical | |
| have hinv : 1 ≤ δ⁻¹ := (one_le_inv₀ hδ).mpr hδ1 | |
| have hceil : (⌈δ⁻¹⌉:ℝ) < δ⁻¹+1 := Int.ceil_lt_add_one _ | |
| have hci : (0:ℤ) ≤ ⌈δ⁻¹⌉ := Int.ceil_nonneg (by positivity) | |
| have hfull : ((fullCapGridND d δ).card : ℝ) = (2*(⌈δ⁻¹⌉:ℝ)+5)^d := by | |
| simp only [fullCapGridND, Fintype.card_piFinset, Finset.prod_const, Finset.card_univ, | |
| Fintype.card_fin, Nat.cast_pow, Int.card_Icc] | |
| have he : ((⌈δ⁻¹⌉+2+1- -(⌈δ⁻¹⌉+2)).toNat : ℝ) = | |
| ((⌈δ⁻¹⌉+2+1- -(⌈δ⁻¹⌉+2) : ℤ):ℝ) := by | |
| exact_mod_cast Int.toNat_of_nonneg (show 0 ≤ ⌈δ⁻¹⌉+2+1- -(⌈δ⁻¹⌉+2) by omega) | |
| rw [he] | |
| push_cast | |
| ring | |
| calc | |
| _ ≤ ((fullCapGridND d δ).card : ℝ) := by | |
| exact_mod_cast Finset.card_filter_le (fullCapGridND d δ) _ | |
| _ = _ := hfull | |
| _ ≤ _ := pow_le_pow_left₀ (by positivity) (by linarith) d | |
| lemma sphereCap_pair_partition {δ : ℝ} (hδ : 0 < δ) | |
| (hs : (d+1:ℝ)*δ ≤ 1/2) (z : EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d)) | |
| (hz : ‖z.1‖=1 ∧ ‖z.2‖=1) : | |
| (∀ k ∈ (sphereCapGridND d δ).product (sphereCapGridND d δ), | |
| 0 ≤ tensorCapND δ k.1 z.1*tensorCapND δ k.2 z.2) ∧ | |
| ∑ k ∈ (sphereCapGridND d δ).product (sphereCapGridND d δ), | |
| tensorCapND δ k.1 z.1*tensorCapND δ k.2 z.2 = 1 := by | |
| classical | |
| constructor | |
| · intro k _; positivity [tensorCapND_nonneg δ k.1 z.1,tensorCapND_nonneg δ k.2 z.2] | |
| · calc | |
| _ = ∑ i ∈ sphereCapGridND d δ, ∑ j ∈ sphereCapGridND d δ, | |
| tensorCapND δ i z.1*tensorCapND δ j z.2 := | |
| Finset.sum_product _ _ _ | |
| _ = (∑ i ∈ sphereCapGridND d δ, tensorCapND δ i z.1)* | |
| (∑ j ∈ sphereCapGridND d δ, tensorCapND δ j z.2) := (Finset.sum_mul_sum _ _ _ _).symm | |
| _ = 1 := by rw [sphereCapGridND_partition hδ hs z.1 hz.1, | |
| sphereCapGridND_partition hδ hs z.2 hz.2,one_mul] | |
| end CurrentFalconer | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace CurrentFalconer | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate ContDiff | |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] | |
| structure GraphSymbolJet (b : V → V → ℂ) (m : ℕ) (s B : ℝ) : Prop where | |
| measurable : Measurable (Function.uncurry b) | |
| smooth : ∀ x, ContDiff ℝ ∞ (b x) | |
| nonneg : 0 ≤ B | |
| bound : ∀ x j, j ≤ m → ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j (b x) w‖ ≤ B*s^j | |
| omit [FiniteDimensional ℝ V] [BorelSpace V] in | |
| lemma GraphSymbolJet.norm_le {b : V → V → ℂ} {m : ℕ} {s B : ℝ} | |
| (h : GraphSymbolJet b m s B) (x w : V) (hw : ‖w‖=1) : ‖b x w‖ ≤ B := by | |
| simpa only [norm_iteratedFDeriv_zero,pow_zero,mul_one] using h.bound x 0 (Nat.zero_le _) w hw | |
| lemma graph_uneligible_transform (ρ σ : Measure V) [IsFiniteMeasure ρ] [IsFiniteMeasure σ] | |
| (P : Set V) (M : ℝ) (η : V → ℝ) (b : V → V → ℂ) | |
| (hη : Measurable η) (hb : Measurable (Function.uncurry b)) | |
| {E B : ℝ} (hE : 0 ≤ E) (hB : 0 ≤ B) | |
| (hη0 : ∀ᵐ w ∂σ, 0 ≤ η w) (hηE : ∀ᵐ w ∂σ, η w ≤ E) | |
| (hbB : ∀ᵐ w ∂σ, ∀ x, ‖b x w‖ ≤ B) | |
| (hz : ¬0 < ∫ x in P, ∫ w, η w*‖b x w‖^2 ∂σ ∂ρ) : | |
| ∀ᵐ w ∂σ, η w=0 ∨ ∀ r, sourceAngularTransformND ρ P M b r w=0 := by | |
| have hz' := zero_cap_energy_transform (ρ.restrict P) σ η b hη hb hE hB hη0 hηE hbB | |
| (le_antisymm (not_lt.mp hz) (cap_energy_nonneg _ _ _ _ hη0)) | |
| filter_upwards [hz'] with w hw | |
| rcases hw with hw | hw | |
| · exact Or.inl hw | |
| right | |
| intro r | |
| have hr := hw r | |
| simp only [sourceAngularTransformND,inv_one,one_smul,Measure.restrict_univ] at hr | |
| simp only [sourceAngularTransformND,hr,smul_zero] | |
| end CurrentFalconer | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace CurrentFalconer.TreeND | |
| open Set MeasureTheory | |
| variable {d : ℕ} | |
| lemma graphChildren_eligible_iff {ρ : Measure (EuclideanSpace ℝ (Fin d))} {a p : ℕ} | |
| {X P : CellPath d} {η : EuclideanSpace ℝ (Fin d) → ℝ} | |
| {b : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ} | |
| (hP : P ∈ graphChildren ρ a p X) : | |
| P ∈ eligibleCells ρ a p X η b ↔ 0 < cellCapEnergy ρ P η b := by | |
| classical | |
| have hP' := Finset.mem_filter.mp (Finset.mem_filter.mp hP).1 | |
| simp only [eligibleCells,Finset.mem_filter,hP'.1,hP'.2,true_and] | |
| end CurrentFalconer.TreeND | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace CurrentFalconer | |
| open Set MeasureTheory Attempt4FullFar | |
| open scoped ComplexConjugate ContDiff | |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] | |
| lemma graphCellLaw_four_ae (ρ₁ ρ₂ ρ₃ ρ₄ : Measure V) | |
| [IsFiniteMeasure ρ₁] [IsFiniteMeasure ρ₂] [IsFiniteMeasure ρ₃] [IsFiniteMeasure ρ₄] | |
| (P₁ P₂ P₃ P₄ : Set V) (M₁ M₂ M₃ M₄ : ℝ) | |
| (hP₁ : MeasurableSet P₁) (hP₂ : MeasurableSet P₂) | |
| (hP₃ : MeasurableSet P₃) (hP₄ : MeasurableSet P₄) : | |
| ∀ᵐ z ∂((graphCellLaw ρ₁ P₁ M₁).prod (graphCellLaw ρ₂ P₂ M₂)).prod | |
| ((graphCellLaw ρ₃ P₃ M₃).prod (graphCellLaw ρ₄ P₄ M₄)), | |
| z.1.1 ∈ P₁ ∧ z.1.2 ∈ P₂ ∧ z.2.1 ∈ P₃ ∧ z.2.2 ∈ P₄ := by | |
| have h₁₂ : ∀ᵐ z ∂(graphCellLaw ρ₁ P₁ M₁).prod (graphCellLaw ρ₂ P₂ M₂), | |
| z.1 ∈ P₁ ∧ z.2 ∈ P₂ := by | |
| filter_upwards [Measure.quasiMeasurePreserving_fst.ae (graphCellLaw_ae ρ₁ hP₁ M₁), | |
| Measure.quasiMeasurePreserving_snd.ae (graphCellLaw_ae ρ₂ hP₂ M₂)] with z h₁ h₂ | |
| exact ⟨h₁,h₂⟩ | |
| have h₃₄ : ∀ᵐ z ∂(graphCellLaw ρ₃ P₃ M₃).prod (graphCellLaw ρ₄ P₄ M₄), | |
| z.1 ∈ P₃ ∧ z.2 ∈ P₄ := by | |
| filter_upwards [Measure.quasiMeasurePreserving_fst.ae (graphCellLaw_ae ρ₃ hP₃ M₃), | |
| Measure.quasiMeasurePreserving_snd.ae (graphCellLaw_ae ρ₄ hP₄ M₄)] with z h₃ h₄ | |
| exact ⟨h₃,h₄⟩ | |
| filter_upwards [Measure.quasiMeasurePreserving_fst.ae h₁₂, | |
| Measure.quasiMeasurePreserving_snd.ae h₃₄] with z h₁₂ h₃₄ | |
| exact ⟨h₁₂.1,h₁₂.2,h₃₄.1,h₃₄.2⟩ | |
| lemma graph_radial_budget {Ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ) | |
| (hs : HasCompactSupport Ψ) (m : ℕ) : | |
| ∃ K : ℝ, 0 ≤ K ∧ (∫ u : ℝ, ‖(Ψ u:ℂ)‖) ≤ K ∧ | |
| ∀ u, ‖iteratedDeriv m (fun u => (Ψ u:ℂ)) u‖ ≤ K := by | |
| let f : ℝ → ℂ := fun u => (Ψ u:ℂ) | |
| have hts : tsupport f = tsupport Ψ := by | |
| change closure (Function.support f) = closure (Function.support Ψ) | |
| congr 1 | |
| ext u | |
| simp only [Function.mem_support, f, ne_eq, Complex.ofReal_eq_zero] | |
| have hs' : HasCompactSupport f := by | |
| change IsCompact (tsupport f) | |
| rw [hts] | |
| exact hs | |
| have hΨ' : ContDiff ℝ ∞ f := Complex.ofRealCLM.contDiff.comp hΨ | |
| obtain ⟨K,hK⟩ := (hs'.iteratedFDeriv m).exists_bound_of_continuous | |
| (hΨ'.continuous_iteratedFDeriv (by exact_mod_cast (le_top : (m:ℕ∞) ≤ ⊤))) | |
| refine ⟨max K (∫ u, ‖f u‖), (integral_nonneg (fun _ => norm_nonneg _)).trans (le_max_right _ _), | |
| le_max_right _ _,?_⟩ | |
| intro u | |
| rw [←norm_iteratedFDeriv_eq_norm_iteratedDeriv] | |
| exact (hK u).trans (le_max_left _ _) | |
| omit [FiniteDimensional ℝ V] [BorelSpace V] in | |
| lemma graph_symbol_pair_budget {b c : V → V → ℂ} {θ : V → ℂ} | |
| {m : ℕ} {s A B C P : ℝ} (hb : GraphSymbolJet b m s B) (hc : GraphSymbolJet c m s C) | |
| (hθ : ContDiff ℝ ∞ θ) (hs : 0 ≤ s) (hsA : s ≤ A) (hP : 0 ≤ P) | |
| (hdθ : ∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j θ w‖ ≤ P*A^j) : | |
| (∀ z, ContDiff ℝ ∞ (graphPairAmplitude θ b c z)) ∧ | |
| ∀ z j, j ≤ m → ∀ w, ‖w‖=1 → | |
| ‖iteratedFDeriv ℝ j (graphPairAmplitude θ b c z) w‖ ≤ | |
| (P*B*2^m*C*2^m)*A^j := by | |
| constructor | |
| · intro z; exact (hθ.mul (hb.smooth z.1)).mul (Complex.conjCLE.contDiff.comp (hc.smooth z.2)) | |
| · intro z j hj w hw | |
| exact graphJetPair hθ (hb.smooth z.1) (hc.smooth z.2) m hP hb.nonneg hc.nonneg | |
| (hs.trans hsA) w (fun k hk => hdθ k hk w hw) | |
| (graphJet_enlarge hb.nonneg hs hsA (fun k hk => hb.bound z.1 k hk w hw)) | |
| (graphJet_enlarge hc.nonneg hs hsA (fun k hk => hc.bound z.2 k hk w hw)) j hj | |
| end CurrentFalconer | |
| end | |
| end | |
| end OAI | |