| import OAI.Geometry.RCD.Sobolev.BoundedTransverseTest | |
| import OAI.Geometry.RCD.Sobolev.TransverseTestFormula | |
| namespace OAI | |
| /-! # The bounded transverse test with its actual, possibly nonorthogonal, Gram entries -/ | |
| open Set Filter MeasureTheory | |
| open scoped NNReal | |
| noncomputable section | |
| namespace RegularPointCoordinates | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] | |
| (μ : Measure X) (Q : QuadraticCheegerEnergy μ) | |
| theorem exists_bounded_transverse_formula (w s η : SobolevSpace μ Q) (Ω : Set X) | |
| (W S : X → ℝ) (A B E : ℝ≥0) | |
| (hw : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ Q w x|≤A) | |
| (hs : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ Q s x|≤B) | |
| (he : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ Q η x|≤E) | |
| (hW : SobolevSpace.toLp μ Q w =ᵐ[μ.restrict Ω] W) | |
| (hS : SobolevSpace.toLp μ Q s =ᵐ[μ.restrict Ω] S) | |
| (hη : SobolevSpace.toLp μ Q η =ᵐ[μ.restrict Ω] fun _ => (1:ℝ)) | |
| (δ c : ℝ) : | |
| ∃ f : SobolevSpace μ Q, | |
| (SobolevSpace.toLp μ Q f =ᵐ[μ.restrict Ω] | |
| fun x => max (W x+δ) 0*max (c-S x) 0) ∧ | |
| (localGamma μ Q f w =ᵐ[μ.restrict Ω] | |
| fun x => (if -δ<W x then max (c-S x) 0*localGamma μ Q w w x else 0) - | |
| (if S x<c then max (W x+δ) 0*localGamma μ Q s w x else 0)) := by | |
| let P := (w+δ • η).upperPositivePart μ Q (le_refl (0:ℝ)) | |
| let T := (c • η-s).upperPositivePart μ Q (le_refl (0:ℝ)) | |
| let C := A+‖δ‖₊*E | |
| let D := B+‖c‖₊*E | |
| have hP : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ Q P x|≤C := bounded_positive_affine_bound μ Q w η A E hw he δ | |
| have hneg : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ Q (-s) x|≤B := by | |
| simp only [map_neg] | |
| filter_upwards [hs,Lp.coeFn_neg (SobolevSpace.toLp μ Q s)] with x hx hy | |
| simpa only [hy,Pi.neg_apply,abs_neg] using hx | |
| have hT : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ Q T x|≤D := by | |
| simpa only [T,D,sub_eq_neg_add,NNReal.coe_add,NNReal.coe_mul] using | |
| bounded_positive_affine_bound μ Q (-s) η B E hneg he c | |
| refine ⟨SobolevSpace.product μ Q P T C D,?_, | |
| transverse_positive_test_formula μ Q w s η Ω W S hW hS hη δ c C D hP hT⟩ | |
| have hU : SobolevSpace.toLp μ Q (w+δ • η) =ᵐ[μ.restrict Ω] fun x => W x+δ := by | |
| simp only [map_add,map_smul] | |
| filter_upwards [hW,hη,ae_restrict_of_ae (Lp.coeFn_add (SobolevSpace.toLp μ Q w) | |
| (δ • SobolevSpace.toLp μ Q η)),ae_restrict_of_ae (Lp.coeFn_smul δ (SobolevSpace.toLp μ Q η))] | |
| with x hx hy ha hm | |
| simp only [ha,Pi.add_apply,hm,Pi.smul_apply,smul_eq_mul,hx,hy,mul_one] | |
| have hV : SobolevSpace.toLp μ Q (c • η-s) =ᵐ[μ.restrict Ω] fun x => c-S x := by | |
| simp only [map_sub,map_smul] | |
| filter_upwards [hS,hη,ae_restrict_of_ae (Lp.coeFn_sub (c • SobolevSpace.toLp μ Q η) | |
| (SobolevSpace.toLp μ Q s)),ae_restrict_of_ae (Lp.coeFn_smul c (SobolevSpace.toLp μ Q η))] | |
| with x hx hy ha hm | |
| simp only [ha,Pi.sub_apply,hm,Pi.smul_apply,smul_eq_mul,hx,hy,mul_one] | |
| filter_upwards [ae_restrict_of_ae (SobolevSpace.product_coe μ Q P T C D hP hT),hU,hV, | |
| ae_restrict_of_ae ((w+δ • η).upperPositivePart_coe μ Q (le_refl (0:ℝ))), | |
| ae_restrict_of_ae ((c • η-s).upperPositivePart_coe μ Q (le_refl (0:ℝ)))] | |
| with x hx hu hv hp ht | |
| rw [hx] | |
| change SobolevSpace.toLp μ Q ((w+δ • η).upperPositivePart μ Q (le_refl (0:ℝ))) x * | |
| SobolevSpace.toLp μ Q ((c • η-s).upperPositivePart μ Q (le_refl (0:ℝ))) x = _ | |
| rw [hp,ht] | |
| simp only [upperLevelPart,hu,hv,sub_zero] | |
| end RegularPointCoordinates | |
| end | |
| end OAI | |