| import OAI.Geometry.NodalSets.Elliptic.RealSquareDifferenceTest |
| import OAI.Geometry.NodalSets.Spectral.SphereEigenDifferentiatedEquation |
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| namespace OAI |
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| namespace Yau.Target |
| open MeasureTheory Yau.Geometry Set |
| open scoped ContDiff |
| noncomputable section |
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| theorem sphere_eigen_square_difference_test (d : SphereEnergyData) (p : Base) |
| (hrho : ContDiff ℝ ∞ (fun x ↦ d.density (sphereChartCoordMap p x))) : |
| ∃ C1 > 0, ∀ (mu : ℝ), mu ≠ 0 → ∀ (f : SphereWeightedL2 d), |
| sphereL2Resolvent d f=mu • f → |
| ∃ H : Fin 4 → Fin 4 → Lp ℝ 2 (volume.restrict (Yau.realCenteredCube 4 (1/2))), |
| (∑ a, ∑ j, ‖H a j‖^2) ≤ C1*(‖f‖^2+‖sphereWeakSolution d f‖^2) ∧ |
| (∀ a j psi, ContDiff ℝ ∞ psi → HasCompactSupport psi → |
| tsupport psi ⊆ Yau.realCenteredCube 4 (1/2) → |
| (∫ x in Yau.realCenteredCube 4 (1/2), |
| (sphereChartDerivativeMap d p a (sphereWeakSolution d f)) x*Yau.coordPartial psi x j) = |
| -(∫ x in Yau.realCenteredCube 4 (1/2), H a j x*psi x)) ∧ |
| ∀ (k : Fin 4) (eta : Yau.Jets.Coord → ℝ), ContDiff ℝ ∞ eta → HasCompactSupport eta → |
| tsupport eta ⊆ interior (Yau.realCenteredCube 4 (3/8)) → |
| ∀ (i : Fin 4) (h : ℝ), |h| ≤ 1/8 → |
| let Q := Yau.realCenteredCube 4 (1/2) |
| let U := Q.indicator (sphereChartDerivativeMap d p k (sphereWeakSolution d f)) |
| let V := fun j ↦ Q.indicator (H k j) |
| let A := fun j ↦ Q.indicator (fun x ↦ ∑ a, sphereChartPrincipalDensity d p x a j*H a k x) |
| let G := fun j ↦ Q.indicator (Yau.realWeakGradientCommutator (sphereChartPrincipalDensity d p) |
| (fun a ↦ sphereChartDerivativeMap d p a (sphereWeakSolution d f)) k j) |
| let F := Q.indicator (sphereEigenForcingDerivative d p mu f k) |
| let v := Yau.realSquareCutoff eta (Yau.realDifferenceQuotient i h U) |
| let P := Yau.realSquareCutoffGradient eta (Yau.realDifferenceQuotient i h U) |
| (fun j ↦ Yau.realDifferenceQuotient i h (V j)) |
| MemLp v 2 volume ∧ (∀ j, MemLp (P j) 2 volume) ∧ |
| (∀ j, Integrable (fun x ↦ Yau.realDifferenceQuotient i h (A j) x*P j x) ∧ |
| Integrable (fun x ↦ Yau.realDifferenceQuotient i h (G j) x*P j x)) ∧ |
| Integrable (fun x ↦ F x*Yau.realDifferenceQuotient i (-h) v x) ∧ |
| (∑ j, ∫ x, Yau.realDifferenceQuotient i h (A j) x*P j x) = |
| -(∫ x, F x*Yau.realDifferenceQuotient i (-h) v x)- |
| (∑ j, ∫ x, Yau.realDifferenceQuotient i h (G j) x*P j x) := by |
| obtain ⟨C1,hC1,hall⟩ := sphere_eigen_differentiated_equation d p hrho |
| refine ⟨C1,hC1,fun mu hmu f heigen ↦ ?_⟩ |
| obtain ⟨H,hbound,hweak,heq⟩ := hall mu hmu f heigen |
| refine ⟨H,hbound,hweak,fun k eta he hc hs i h hh ↦ ?_⟩ |
| dsimp only |
| let Q := Yau.realCenteredCube 4 (1/2) |
| have hQ : IsCompact Q := Yau.realCenteredCube_isCompact 4 (1/2) |
| have hsub : Q ⊆ realFinCube 4 := Yau.realCenteredCube_mono (by norm_num) |
| let U : Yau.Jets.Coord → ℝ := Q.indicator (sphereChartDerivativeMap d p k (sphereWeakSolution d f)) |
| let V : Fin 4 → Yau.Jets.Coord → ℝ := fun j ↦ Q.indicator (H k j) |
| let A : Fin 4 → Yau.Jets.Coord → ℝ := fun j ↦ Q.indicator |
| (fun x ↦ ∑ a, sphereChartPrincipalDensity d p x a j*H a k x) |
| let G : Fin 4 → Yau.Jets.Coord → ℝ := fun j ↦ Q.indicator |
| (Yau.realWeakGradientCommutator (sphereChartPrincipalDensity d p) |
| (fun a ↦ sphereChartDerivativeMap d p a (sphereWeakSolution d f)) k j) |
| let F : Yau.Jets.Coord → ℝ := Q.indicator (sphereEigenForcingDerivative d p mu f k) |
| have hU : MemLp U 2 volume := (memLp_indicator_iff_restrict hQ.measurableSet).mpr |
| ((Lp.memLp _).mono_measure (Measure.restrict_mono hsub le_rfl)) |
| have hV (j : Fin 4) : MemLp (V j) 2 volume := |
| (memLp_indicator_iff_restrict hQ.measurableSet).mpr (Lp.memLp (H k j)) |
| have hA (j : Fin 4) : MemLp (A j) 2 volume := by |
| apply (memLp_indicator_iff_restrict hQ.measurableSet).mpr |
| apply memLp_finsetSum |
| intro a _ |
| obtain ⟨_,_,hb⟩ := Yau.real_compact_multiplier_bound hQ _ |
| (sphereChartPrincipalDensity_smooth d p a j).continuous |
| exact (hb (H a k) (Lp.memLp _)).1 |
| have hG (j : Fin 4) : MemLp (G j) 2 volume := |
| (memLp_indicator_iff_restrict hQ.measurableSet).mpr ((heq k).2.1 j j).1 |
| have hF : MemLp F 2 volume := (memLp_indicator_iff_restrict hQ.measurableSet).mpr (heq k).1 |
| have hzpair (a b : Yau.Jets.Coord → ℝ) : |
| (∫ x, Q.indicator a x*b x) = ∫ x in Q, a x*b x := by |
| simp only [← indicator_mul_left,integral_indicator hQ.measurableSet] |
| have hW (j : Fin 4) (psi : Yau.Jets.Coord → ℝ) (hp : ContDiff ℝ ∞ psi) |
| (hpc : HasCompactSupport psi) (hps : tsupport psi ⊆ Q) : |
| (∫ x, U x*Yau.coordPartial psi x j)=-(∫ x, V j x*psi x) := by |
| change (∫ x, Q.indicator (fun y ↦ (sphereChartDerivativeMap d p k (sphereWeakSolution d f)) y) x* |
| Yau.coordPartial psi x j)=-(∫ x, Q.indicator (fun y ↦ H k j y) x*psi x) |
| rw [hzpair,hzpair] |
| exact hweak k j psi hp hpc hps |
| have hE (psi : Yau.Jets.Coord → ℝ) (hp : ContDiff ℝ ∞ psi) |
| (hpc : HasCompactSupport psi) (hps : tsupport psi ⊆ Q) : |
| (∑ j, ∫ x, A j x*Yau.coordPartial psi x j) = |
| (∫ x, F x*psi x)-(∑ j, ∫ x, G j x*Yau.coordPartial psi x j) := by |
| have hx := (heq k).2.2 psi hp hpc hps |
| have hsum (j : Fin 4) : (∫ x, A j x*Yau.coordPartial psi x j) = ∑ a, ∫ x in Q, |
| sphereChartPrincipalDensity d p x a j*H a k x*Yau.coordPartial psi x j := by |
| change (∫ x, Q.indicator (fun y ↦ ∑ a, sphereChartPrincipalDensity d p y a j*H a k y) x* |
| Yau.coordPartial psi x j) = _ |
| rw [hzpair] |
| simp only [Finset.sum_mul] |
| exact integral_finsetSum Finset.univ (fun a _ ↦ hx.1 a j) |
| simp only [hsum] |
| rw [Finset.sum_comm] |
| change (∑ a, ∑ j, ∫ x in Q, sphereChartPrincipalDensity d p x a j*H a k x* |
| Yau.coordPartial psi x j) = (∫ x, Q.indicator (sphereEigenForcingDerivative d p mu f k) x*psi x)- |
| (∑ j, ∫ x, Q.indicator (Yau.realWeakGradientCommutator (sphereChartPrincipalDensity d p) |
| (fun a ↦ sphereChartDerivativeMap d p a (sphereWeakSolution d f)) k j) x*Yau.coordPartial psi x j) |
| simp only [hzpair] |
| exact hx.2.2.2 |
| exact Yau.real_square_cutoff_difference_test A G F hA hG hF (1/2) (3/8) (by norm_num) |
| hE U V hU hV hW eta he hc hs i h (by linarith) |
|
|
| end |
| end Yau.Target |
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| end OAI |
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