| import OAI.Geometry.RCD.Heat.BoundedHeatMixedGram | |
| import OAI.Geometry.RCD.Heat.BoundedHeatSobolevData | |
| import OAI.Geometry.RCD.Heat.HeatIteratedLaplacian | |
| import OAI.Geometry.RCD.Heat.HeatBridgeSecondDerivative | |
| import OAI.Geometry.RCD.Sobolev.SobolevSecondVariation | |
| namespace OAI | |
| /-! # The actual bounded heat bridge has the exact Hessian second variation | |
| All spatial representatives, mixed Gram entries, and iterated sources are | |
| constructed from the three original bounded L2 data. The spatial test is | |
| a positive-time heat regularization. No second-order premise is assumed. | |
| -/ | |
| open Set Filter MeasureTheory TopologicalSpace | |
| open scoped ENNReal NNReal Topology InnerProductSpace | |
| noncomputable section | |
| namespace RegularPointCoordinates.NoncollapsedRCD | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] | |
| [ProperSpace X] {K : ℝ} {n : ℕ} (hX : NoncollapsedRCD K n X) | |
| [(hausdorff n X).Regular] [SeparableSpace X] | |
| [IsLocallyFiniteMeasure (hausdorff n X)] (hn : 2 ≤ n) (p : X) | |
| include hn p in | |
| theorem bounded_heat_second_variation (f g h : Lp ℝ 2 (hausdorff n X)) | |
| (A B C : ℝ≥0) (hf : ∀ᵐ x ∂hausdorff n X, |f x| ≤ A) | |
| (hg : ∀ᵐ x ∂hausdorff n X, |g x| ≤ B) (hh : ∀ᵐ x ∂hausdorff n X, |h x| ≤ C) | |
| {T t a : ℝ} (ht : 0 < t) (htT : t < T) (ha : 0 < a) : | |
| let μ := hausdorff n X | |
| let Q := hX.quadratic | |
| let u := heatSobolev μ Q t ht f | |
| let v := heatSobolev μ Q (T-t) (sub_pos.mpr htT) g | |
| let w := heatSobolev μ Q a ha h | |
| ∃ (W : Lp ℝ ∞ μ) (uv wu wv : SobolevSpace μ Q), | |
| W =ᵐ[μ] SobolevSpace.toLp μ Q w ∧ | |
| SobolevSpace.toLp μ Q uv =ᵐ[μ] localGamma μ Q u v ∧ | |
| SobolevSpace.toLp μ Q wu =ᵐ[μ] localGamma μ Q w u ∧ | |
| SobolevSpace.toLp μ Q wv =ᵐ[μ] localGamma μ Q w v ∧ | |
| HasDerivAt (deriv (fun s => ∫ x, W x* | |
| (kernelHeat μ Q (hX.canonicalHeatKernel_isContinuousHeatKernel hn p) s f x* | |
| kernelHeat μ Q (hX.canonicalHeatKernel_isContinuousHeatKernel hn p) (T-s) g x) ∂μ)) | |
| (inner ℝ (heatSecondGeneratorAt μ Q a h) | |
| (SobolevSpace.toLp μ Q (SobolevSpace.product μ Q u v A B))- | |
| 4*(∫ x, localHessianAction μ Q w u v wu wv uv x ∂μ)) t := by | |
| intro μ Q u v w | |
| obtain ⟨huB,Lu,huL⟩ := hX.bounded_heat_sobolev_data hn p f A hf ht | |
| obtain ⟨hvB,Lv,hvL⟩ := hX.bounded_heat_sobolev_data hn p g B hg (sub_pos.mpr htT) | |
| obtain ⟨hwB,-⟩ := hX.bounded_heat_sobolev_data hn p h C hh ha | |
| have hUm : MemLp (SobolevSpace.toLp μ Q u) ∞ μ := | |
| memLp_top_of_bound (Lp.aestronglyMeasurable _) A (by simpa only [Real.norm_eq_abs] using huB) | |
| have hVm : MemLp (SobolevSpace.toLp μ Q v) ∞ μ := | |
| memLp_top_of_bound (Lp.aestronglyMeasurable _) B (by simpa only [Real.norm_eq_abs] using hvB) | |
| have hWm : MemLp (SobolevSpace.toLp μ Q w) ∞ μ := | |
| memLp_top_of_bound (Lp.aestronglyMeasurable _) C (by simpa only [Real.norm_eq_abs] using hwB) | |
| let U := hUm.toLp _ | |
| let V := hVm.toLp _ | |
| let W := hWm.toLp _ | |
| obtain ⟨uv,huv⟩ := hX.bounded_heat_mixed_gram hn p f g A B hf hg ht (sub_pos.mpr htT) | |
| obtain ⟨wu,hwu⟩ := hX.bounded_heat_mixed_gram hn p h f C A hh hf ha ht | |
| obtain ⟨wv,hwv⟩ := hX.bounded_heat_mixed_gram hn p h g C B hh hg ha (sub_pos.mpr htT) | |
| obtain ⟨du,hduv,hdu⟩ := heat_generator_iterated_laplacian μ Q f ht | |
| obtain ⟨dv,hdvv,hdv⟩ := heat_generator_iterated_laplacian μ Q g (sub_pos.mpr htT) | |
| obtain ⟨dw,hdwv,hdw⟩ := heat_generator_iterated_laplacian μ Q h ha | |
| have hu : IsWeakLaplacian μ Q u (-heatGeneratorAt μ Q t f) := by | |
| simpa only [heatGeneratorAt_of_pos μ Q t ht] using heatSobolev_weak_laplacian μ Q t ht f | |
| have hv : IsWeakLaplacian μ Q v (-heatGeneratorAt μ Q (T-t) g) := by | |
| simpa only [heatGeneratorAt_of_pos μ Q (T-t) (sub_pos.mpr htT)] using | |
| heatSobolev_weak_laplacian μ Q (T-t) (sub_pos.mpr htT) g | |
| have hw : IsWeakLaplacian μ Q w (-heatGeneratorAt μ Q a h) := by | |
| simpa only [heatGeneratorAt_of_pos μ Q a ha] using heatSobolev_weak_laplacian μ Q a ha h | |
| have hid := sobolev_second_variation_hessian μ Q u v w du dv dw uv wu wv | |
| (-heatGeneratorAt μ Q t f) (-heatGeneratorAt μ Q (T-t) g) (-heatGeneratorAt μ Q a h) | |
| (heatSecondGeneratorAt μ Q t f) (heatSecondGeneratorAt μ Q (T-t) g) | |
| (heatSecondGeneratorAt μ Q a h) hu hv hw hdu hdv hdw hduv hdvv hdwv U V W | |
| hUm.coeFn_toLp hVm.coeFn_toLp hWm.coeFn_toLp huv hwu hwv A B C Lu Lv huB hvB hwB huL hvL | |
| have heq : heatBridgeSecondVariation μ Q W f g T t = | |
| inner ℝ (heatSecondGeneratorAt μ Q a h) | |
| (SobolevSpace.toLp μ Q (SobolevSpace.product μ Q u v A B))- | |
| 4*(∫ x, localHessianAction μ Q w u v wu wv uv x ∂μ) := by | |
| rw [← hid] | |
| simp only [heatBridgeSecondVariation,u,v,heatSobolev_toLp,map_neg, | |
| inner_neg_left,inner_neg_right,neg_neg] | |
| ring | |
| refine ⟨W,uv,wu,wv,hWm.coeFn_toLp,huv,hwu,hwv,?_⟩ | |
| rw [← heq] | |
| exact kernelHeat_bridge_pairing_second_hasDerivAt μ Q | |
| (hX.canonicalHeatKernel_isContinuousHeatKernel hn p) W f g ht htT | |
| end RegularPointCoordinates.NoncollapsedRCD | |
| end | |
| end OAI | |