File size: 3,677 Bytes
9eeb9c7 | 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 68 69 70 71 72 73 74 75 76 77 78 79 | import OAI.Analysis.HyperbolicCones.KernelLocal
import OAI.Analysis.HyperbolicCones.Coordinates
namespace OAI
noncomputable section
open scoped ContDiff Matrix.Norms.L2Operator
open Set Matrix ContinuousLinearMap
namespace Paper256
theorem contDiff_outerSym : ContDiff ℝ ∞ (outerSym : Vec 4 → Sym 4) := by
have hcoord : ContDiff ℝ ∞ (fun v : Vec 4 =>
fun ij : SymIndex => v ij.val.1 * v ij.val.2) := by
apply contDiff_pi.mpr
intro ij
exact (contDiff_piLp_apply (𝕜 := ℝ) (p := 2) (i := ij.val.1)).mul
(contDiff_piLp_apply (𝕜 := ℝ) (p := 2) (i := ij.val.2))
have heq : (outerSym : Vec 4 → Sym 4) = fun v =>
symmetricCoordinateEquiv.symm (fun ij => v ij.val.1 * v ij.val.2) := by
funext v
apply symmetricCoordinateEquiv.injective
rw [LinearEquiv.apply_symm_apply]
rfl
rw [heq]
exact symmetricCoordinateEquiv.symm.toContinuousLinearEquiv.contDiff.comp hcoord
def symmetricOperatorLinear (c : ℕ) : Sym c →ₗ[ℝ] (Vec c →L[ℝ] Vec c) :=
(Matrix.toEuclideanCLM (n := Fin c) (𝕜 := ℝ)).toAlgEquiv.toLinearEquiv.toLinearMap.comp
(selfAdjoint.submodule ℝ (Mat c ℝ)).subtype
theorem contDiff_kernelFamily (c : ℕ) (E : Sym 4 →ₗ[ℝ] Sym c) :
ContDiff ℝ ∞ (fun u => symmetricOperatorLinear c (E (1 - outerSym u))) := by
exact (symmetricOperatorLinear c).toContinuousLinearMap.contDiff.comp
(E.toContinuousLinearMap.contDiff.comp (contDiff_const.sub contDiff_outerSym))
theorem kernelFamily_isSelfAdjoint (c : ℕ) (E : Sym 4 →ₗ[ℝ] Sym c) (u : Vec 4) :
IsSelfAdjoint (symmetricOperatorLinear c (E (1 - outerSym u))) :=
(E (1 - outerSym u)).property.map (Matrix.toEuclideanCLM (n := Fin c) (𝕜 := ℝ))
theorem kernel_projection_smooth_local_extension (c : ℕ) (_hC : 0 < c)
(E : Sym 4 →ₗ[ℝ] Sym c) :
∃ (v : Vec 4) (U : Set (Vec 4)) (F : Vec 4 → Mat c ℝ),
‖v‖ = 1 ∧ v 0 ≠ 0 ∧ IsOpen U ∧ v ∈ U ∧ ContDiffOn ℝ ∞ F U ∧
∀ u ∈ U, ‖u‖ = 1 →
F u = kernelProjection (E (1 - outerSym u) : Mat c ℝ) := by
let A : Vec 4 → Vec c →L[ℝ] Vec c := fun u => symmetricOperatorLinear c (E (1 - outerSym u))
have hA : ContDiff ℝ ∞ A := contDiff_kernelFamily c E
let S : Set (Vec 4) := {u | ‖u‖ = 1 ∧ u 0 ≠ 0}
have hs : S.Nonempty := ⟨PiLp.single 2 0 1, by simp [S]⟩
have hself (u : Vec 4) (_hu : u ∈ S) : IsSelfAdjoint (A u) :=
kernelFamily_isSelfAdjoint c E u
obtain ⟨v, hv, U, P, hU, hvU, hP, heq⟩ := exists_smooth_kernel_extension S hs A hA hself
let U' := U ∩ {u : Vec 4 | u 0 ≠ 0}
let F : Vec 4 → Mat c ℝ := fun u =>
(Matrix.toEuclideanCLM (n := Fin c) (𝕜 := ℝ)).symm (P u)
refine ⟨v, U', F, hv.1, hv.2, ?_, ⟨hvU, hv.2⟩, ?_, ?_⟩
· have hc : Continuous (fun u : Vec 4 => u 0) :=
(contDiff_piLp_apply (𝕜 := ℝ) (p := 2) (n := ∞) (i := (0 : Fin 4))).continuous
exact hU.inter (isOpen_ne.preimage hc)
· exact (Matrix.toEuclideanCLM (n := Fin c) (𝕜 := ℝ)).symm.toAlgEquiv.toLinearEquiv.toContinuousLinearEquiv.contDiff.comp_contDiffOn
(hP.mono (fun _ hu => hu.1))
· intro u hu hunit
apply (Matrix.toEuclideanCLM (n := Fin c) (𝕜 := ℝ)).injective
change (Matrix.toEuclideanCLM (n := Fin c) (𝕜 := ℝ))
((Matrix.toEuclideanCLM (n := Fin c) (𝕜 := ℝ)).symm (P u)) = _
rw [StarAlgEquiv.apply_symm_apply, heq u hu.1 ⟨hunit, hu.2⟩]
apply ContinuousLinearMap.coe_injective
change _ = Matrix.toEuclideanLin (kernelProjection (E (1 - outerSym u) : Mat c ℝ))
rw [kernelProjection, LinearEquiv.apply_symm_apply]
rfl
end Paper256
end
end OAI
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