| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Slicing.NormalRestriction |
| import OAI.Geometry.CAT0Fillings.Gradient.HilbertAtlas |
| import OAI.Geometry.CAT0Fillings.Mass.Convolution |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology NNReal ENNReal BigOperators |
|
|
| namespace CAT0Fillings |
| namespace ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| theorem ae_coarea_mass_le_gradient {u : X → ℝ} {K : ℝ≥0} |
| (hu : LipschitzWith K u) {S : ℝ → Functional X k} |
| (hS : ∀ᵐ t : ℝ, IsMetricCurrent (S t)) |
| (hact : ∀ b π, Admissible b π → Integrable (fun t : ℝ => S t b π) ∧ |
| T b (Matrix.vecCons u π) = ∫ t : ℝ, S t b π) |
| (hweight : ∀ b π, Admissible b π → ∀ φ : ℝ → ℝ, BoundedLip (φ ∘ u) → |
| ∀ᵐ t : ℝ, S t (fun x => φ (u x)*b x) π = φ t * S t b π) : |
| ∀ᵐ t : ℝ, mass (S t) ≤ (((q.gradientMeasure u).map u).rnDeriv volume t).toReal := by |
| let := q.gradientMeasure_finite hu |
| have hub : BoundedLip u := ⟨⟨K,hu⟩,by |
| obtain ⟨M,hM⟩ := isCompact_univ.exists_bound_of_continuousOn hu.continuous.continuousOn |
| exact ⟨M,fun x => by simpa only [Real.norm_eq_abs] using hM x (mem_univ _)⟩⟩ |
| apply BorelCoefficients.ae_mass_le_of_listwise volume hS |
| intro N b π hb hbudget hπ |
| have hab i : Admissible (b i) (π i) := ⟨hb i,fun j => ⟨1,hπ i j⟩⟩ |
| have hi : Integrable (fun t => ∑ i, S t (b i) (π i)) := |
| integrable_finsetSum _ (fun i _ => (hact _ _ (hab i)).1) |
| filter_upwards [SmoothCutoff.ae_tendsto_dprofile_integral hi, |
| SmoothCutoff.ae_tendsto_dprofile_measure ((q.gradientMeasure u).map u)] with t ht hγ |
| have hle (n : ℕ) : (∫ s : ℝ, SmoothCutoff.dprofile ((n:ℝ≥0)+1) t s * |
| (∑ i, S s (b i) (π i))) ≤ |
| ∫ s, SmoothCutoff.dprofile ((n:ℝ≥0)+1) t s ∂(q.gradientMeasure u).map u := by |
| let φ := SmoothCutoff.dprofile ((n:ℝ≥0)+1) t |
| have hφ : BoundedLip (φ ∘ u) := Slicing.dprofile_comp_boundedLip hub _ t |
| have hφ0 x : 0 ≤ φ x := mul_nonneg (NNReal.coe_nonneg _) |
| Real.smoothTransition.monotone.deriv_nonneg |
| have heq i : T (fun x => φ (u x)*b i x) (Fin.cons u (π i)) = |
| ∫ s : ℝ, φ s * S s (b i) (π i) := by |
| change T (fun x => (φ ∘ u) x*b i x) (Matrix.vecCons u (π i)) = _ |
| rw [(hact _ _ ⟨hφ.mul (hb i),(hab i).2⟩).2] |
| exact integral_congr_ae (hweight _ _ (hab i) φ hφ) |
| calc |
| _ = ∑ i, ∫ s : ℝ, φ s * S s (b i) (π i) := by |
| simp_rw [Finset.mul_sum] |
| exact integral_finsetSum _ (fun i _ => SmoothCutoff.dprofile_mul_integrable (hact _ _ (hab i)).1 _ _) |
| _ = ∑ i, T (fun x => φ (u x)*b i x) (Fin.cons u (π i)) := by simp_rw [heq] |
| _ ≤ |∑ i, T (fun x => φ (u x)*b i x) (Fin.cons u (π i))| := le_abs_self _ |
| _ ≤ ∫ x, |φ (u x)| ∂q.gradientMeasure u := |
| q.list_action_le_gradient hu hφ b hb hbudget π hπ |
| _ = ∫ x, φ (u x) ∂q.gradientMeasure u := by |
| simp only [abs_of_nonneg (hφ0 _)] |
| _ = _ := (integral_map hu.continuous.measurable.aemeasurable |
| (SmoothCutoff.dprofile_integrable_measure ((q.gradientMeasure u).map u) _ _).aestronglyMeasurable).symm |
| exact le_of_tendsto_of_tendsto ht hγ (Eventually.of_forall hle) |
|
|
| end ChartGeometry |
| end CAT0Fillings |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology NNReal ENNReal BigOperators |
|
|
| namespace CAT0Fillings |
|
|
| lemma densityPush_add_of_nonneg {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| {μ : Measure α} {φ : α → β} (hφ : Measurable φ) {ρ σ : α → ℝ} |
| (hρm : AEMeasurable ρ μ) (hρ : 0 ≤ᵐ[μ] ρ) (hσ : 0 ≤ᵐ[μ] σ) : |
| densityPush μ φ (fun z => ρ z+σ z) = densityPush μ φ ρ + densityPush μ φ σ := by |
| have hwd : μ.withDensity (fun z => ENNReal.ofReal (ρ z+σ z)) = |
| μ.withDensity (fun z => ENNReal.ofReal (ρ z)) + |
| μ.withDensity (fun z => ENNReal.ofReal (σ z)) := by |
| apply Measure.ext |
| intro s hs |
| simp only [withDensity_apply _ hs,Measure.add_apply] |
| calc |
| _ = ∫⁻ z in s, ENNReal.ofReal (ρ z)+ENNReal.ofReal (σ z) ∂μ := by |
| apply lintegral_congr_ae |
| filter_upwards [ae_restrict_of_ae hρ,ae_restrict_of_ae hσ] with z hz hzs |
| exact ENNReal.ofReal_add hz hzs |
| _ = _ := lintegral_add_left' (hρm.ennreal_ofReal.restrict) _ |
| simp only [densityPush,hwd,Measure.map_add _ _ hφ] |
| |
| lemma weighted_young_real (c ρ d e : ℝ) (hρ : 0 ≤ ρ) (he : d^2 = e) : |
| (2*c)*(ρ*d) ≤ c^2*ρ+ρ*e := by |
| rw [←he] |
| nlinarith [mul_nonneg hρ (sq_nonneg (c-d))] |
| |
| lemma densityPush_quadratic_le {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| {μ : Measure α} {φ : α → β} (hφ : Measurable φ) (ρ d e : α → ℝ) |
| (hρm : AEMeasurable ρ μ) (hρ : ∀ z, 0 ≤ ρ z) |
| (hd : ∀ᵐ z ∂μ, d z^2 = e z) (c : ℝ≥0) : |
| (2*c) • densityPush μ φ (fun z => ρ z*d z) ≤ |
| c^2 • densityPush μ φ ρ + densityPush μ φ (fun z => ρ z*e z) := by |
| have hpoint : (fun z => (2*(c:ℝ))*(ρ z*d z)) ≤ᵐ[μ] |
| (fun z => (c:ℝ)^2*ρ z + ρ z*e z) := |
| hd.mono fun z hz => weighted_young_real _ _ _ _ (hρ z) hz |
| have hi := densityPush_mono_ae hφ hpoint |
| have hadd : densityPush μ φ (fun z => (c:ℝ)^2*ρ z + ρ z*e z) = |
| densityPush μ φ (fun z => (c:ℝ)^2*ρ z) + densityPush μ φ (fun z => ρ z*e z) := |
| densityPush_add_of_nonneg hφ (hρm.const_mul _) |
| (Eventually.of_forall fun z => mul_nonneg (sq_nonneg _) (hρ z)) |
| (hd.mono fun z hz => mul_nonneg (hρ z) (hz ▸ sq_nonneg (d z))) |
| rw [hadd] at hi |
| have hl : densityPush μ φ (fun z => (2*(c:ℝ))*(ρ z*d z)) = |
| (2*c) • densityPush μ φ (fun z => ρ z*d z) := by |
| simpa only [NNReal.coe_mul, NNReal.coe_ofNat] using |
| densityPush_const_mul μ φ (fun z => ρ z*d z) (2*c) hφ |
| have hr : densityPush μ φ (fun z => (c:ℝ)^2*ρ z) = |
| c^2 • densityPush μ φ ρ := by |
| simpa only [NNReal.coe_pow] using densityPush_const_mul μ φ ρ (c^(2:ℕ)) hφ |
| rwa [hl,hr] at hi |
| |
| namespace ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X k} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
| |
| lemma chart_gradient_quadratic_le (i : ℕ) {u : X → ℝ} {K : ℝ≥0} |
| (hu : LipschitzWith K u) (c : ℝ≥0) : |
| (2*c) • densityPush (volume.restrict (q.chart i).domain) (q.chart i).paramExtended |
| (fun z => q.density i z*q.duNorm i u z) ≤ |
| c^2 • densityPush (volume.restrict (q.chart i).domain) (q.chart i).paramExtended |
| (q.density i) + |
| densityPush (volume.restrict (q.chart i).domain) (q.chart i).paramExtended |
| (fun z => q.density i z*q.normSq i u z) := |
| densityPush_quadratic_le (q.chart i).measurable_paramExtended _ _ _ |
| (q.density_integrable i).aestronglyMeasurable.aemeasurable (q.density_nonneg i) |
| ((q.ae_duNorm_bounds i hu).mono fun _ hz => hz.2.2) c |
| |
| lemma gradient_quadratic_le {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) |
| (c : ℝ≥0) : (2*c) • q.gradientMeasure u ≤ |
| c^2 • MassMeasure.currentMassMeasure hT + q.energyMeasure u := by |
| apply Measure.le_iff.mpr |
| intro s hs |
| rw [Measure.add_apply,Measure.smul_apply,Measure.smul_apply,q.measure_eq] |
| simp only [gradientMeasure,energyMeasure,weightedMeasure,Measure.sum_apply _ hs] |
| change (2*(c:ℝ≥0∞)) * _ ≤ (c:ℝ≥0∞)^2 * _ + _ |
| rw [←ENNReal.tsum_mul_left,←ENNReal.tsum_mul_left,←ENNReal.tsum_add] |
| apply ENNReal.tsum_le_tsum |
| intro i |
| exact q.chart_gradient_quadratic_le i hu c s |
| |
| end ChartGeometry |
| end CAT0Fillings |
| end |
| |
| end OAI |
| |
