| import OAI.Analysis.HyperbolicCones.KernelLocal |
| import OAI.Analysis.HyperbolicCones.Coordinates |
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|
| namespace OAI |
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| noncomputable section |
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|
| open scoped ContDiff Matrix.Norms.L2Operator |
| open Set Matrix ContinuousLinearMap |
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| namespace Paper256 |
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| 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 |
|
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| 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 |
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| 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)) |
|
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| 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)) |
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| 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 |
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| end Paper256 |
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| end |
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| end OAI |
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|
