import OAI.Dynamics.TriangleBilliards.SecondDerivatives namespace OAI universe uA open MeasureTheory Set open scoped ENNReal symmDiff noncomputable section open MeasureTheory Set Filter Function Metric open scoped Topology Convolution ContDiff noncomputable section open MeasureTheory Set open scoped ENNReal noncomputable section open MeasureTheory Set Filter BoundedContinuousFunction open scoped ENNReal Topology ComplexConjugate noncomputable section open MeasureTheory Set Filter open scoped Topology ComplexConjugate noncomputable section open MeasureTheory Filter open scoped ComplexConjugate noncomputable section open MeasureTheory Filter Set open scoped Topology ComplexConjugate noncomputable section open Filter Finset Set open scoped Topology BigOperators noncomputable section open MeasureTheory Filter Set open scoped Topology ContDiff NNReal open MeasureTheory Filter Set open scoped Topology ComplexConjugate noncomputable section namespace TriangularBilliards.Analysis variable {A : Type uA} [MeasurableSpace A] {μ : Measure A} def restrictL2 (E : Set A) (hE : MeasurableSet E) : Lp ℂ 2 μ →L[ℂ] Lp ℂ 2 μ := LinearMap.mkContinuous { toFun := fun u => ((Lp.memLp u).indicator hE).toLp (E.indicator u) map_add' := fun u v => by apply Lp.ext filter_upwards [((Lp.memLp (u+v)).indicator hE).coeFn_toLp, ((Lp.memLp u).indicator hE).coeFn_toLp, ((Lp.memLp v).indicator hE).coeFn_toLp, Lp.coeFn_add u v, Lp.coeFn_add (((Lp.memLp u).indicator hE).toLp _) (((Lp.memLp v).indicator hE).toLp _)] with z h0 h1 h2 h3 h4 simp only [Pi.add_apply] at h3 h4 rw [h0, h4, h1, h2] by_cases hz : z ∈ E · simp only [indicator_of_mem hz]; exact h3 · simp only [indicator_of_notMem hz, add_zero] map_smul' := fun c u => by apply Lp.ext filter_upwards [((Lp.memLp (c • u)).indicator hE).coeFn_toLp, ((Lp.memLp u).indicator hE).coeFn_toLp, Lp.coeFn_smul c u, Lp.coeFn_smul c (((Lp.memLp u).indicator hE).toLp _)] with z h0 h1 h2 h3 simp only [Pi.smul_apply] at h2 h3 change _ = (c • (((Lp.memLp u).indicator hE).toLp _)) z rw [h0, h3, h1] by_cases hz : z ∈ E · simp only [indicator_of_mem hz]; exact h2 · simp only [indicator_of_notMem hz, smul_zero] } 1 (fun u => by simp only [one_mul] change ‖((Lp.memLp u).indicator hE).toLp _‖ ≤ ‖u‖ apply Lp.norm_le_norm_of_ae_le filter_upwards [((Lp.memLp u).indicator hE).coeFn_toLp] with z hz rw [hz] exact norm_indicator_le_norm_self _ _) lemma restrictL2_coe (E : Set A) (hE : MeasurableSet E) (u : Lp ℂ 2 μ) : (restrictL2 E hE u : A → ℂ) =ᵐ[μ] E.indicator u := ((Lp.memLp u).indicator hE).coeFn_toLp lemma restrictL2_selfadjoint (E : Set A) (hE : MeasurableSet E) (u v : Lp ℂ 2 μ) : inner ℂ (restrictL2 E hE u) v = inner ℂ u (restrictL2 E hE v) := by rw [L2.inner_def, L2.inner_def] apply integral_congr_ae filter_upwards [restrictL2_coe E hE u, restrictL2_coe E hE v] with z hu hv rw [hu, hv] by_cases hz : z ∈ E · simp only [indicator_of_mem hz] · simp only [indicator_of_notMem hz, inner_zero_left, inner_zero_right] lemma restrictL2_fixed (E : Set A) (hE : MeasurableSet E) (u : Lp ℂ 2 μ) (hu : ∀ᵐ z ∂μ, z ∉ E → u z = 0) : restrictL2 E hE u = u := by apply Lp.ext filter_upwards [restrictL2_coe E hE u, hu] with z hz hzz rw [hz] by_cases hm : z ∈ E · exact indicator_of_mem hm _ · simp only [indicator_of_notMem hm, hzz hm] lemma restrictL2_commute (E : Set A) (hE : MeasurableSet E) (ρ : A → A) (hρ : MeasurePreserving ρ μ μ) (he : ∀ z, ρ z ∈ E ↔ z ∈ E) (u : Lp ℂ 2 μ) : restrictL2 E hE (Lp.compMeasurePreserving ρ hρ u) = Lp.compMeasurePreserving ρ hρ (restrictL2 E hE u) := by apply Lp.ext filter_upwards [restrictL2_coe E hE (Lp.compMeasurePreserving ρ hρ u), Lp.coeFn_compMeasurePreserving u hρ, Lp.coeFn_compMeasurePreserving (restrictL2 E hE u) hρ, hρ.quasiMeasurePreserving.ae (restrictL2_coe E hE u)] with z h0 h1 h2 h3 rw [h0, h2] change _ = restrictL2 E hE u (ρ z) rw [h3] by_cases hz : z ∈ E · rw [indicator_of_mem hz, indicator_of_mem ((he z).mpr hz)] exact h1 · rw [indicator_of_notMem hz, indicator_of_notMem (fun h => hz ((he z).mp h))] lemma restrictL2_norm (E : Set A) (hE : MeasurableSet E) (u : Lp ℂ 2 μ) : ‖restrictL2 E hE u‖ ≤ ‖u‖ := by apply Lp.norm_le_norm_of_ae_le filter_upwards [restrictL2_coe E hE u] with z hz rw [hz] exact norm_indicator_le_norm_self _ _ end TriangularBilliards.Analysis namespace TriangularBilliards open Analysis MeasureTheory Filter Set open scoped ContDiff NNReal local instance : Fact (0 < 2 * Real.pi) := ⟨by positivity⟩ lemma spatialRestriction_commute (Q : Triangle) (E : Set ℂ) (hE : MeasurableSet E) (θ : AddCircle (2 * Real.pi)) (u : Lp ℂ 2 (doubleMeasure Q)) : restrictL2 {z : DoublePhase | z.1.1 ∈ E} (hE.preimage measurable_fst.fst) ((angularCircleAction Q).act θ u) = (angularCircleAction Q).act θ (restrictL2 {z : DoublePhase | z.1.1 ∈ E} (hE.preimage measurable_fst.fst) u) := by exact restrictL2_commute _ _ (doubleRotate θ.toCircle) (measurePreserving_doubleRotate Q θ.toCircle) (fun _ => Iff.rfl) u /-- The localized energy bound on the actual smooth core. Every generator and Fourier projection here is that of the actual billiard double. -/ lemma seam_localized_energy {Q : Triangle} {f : DoublePhase → ℂ} (hs : SeamCompatible Q f) (hm : Measurable f) (hd : ∀ v b, ContDiff ℝ ∞ (fun x => f ((x,v),b))) {C D : ℝ≥0} (hC : ∀ x v b, ‖fderiv ℝ (fun y => f ((y,v),b)) x‖ ≤ C) (hD : ∀ x v b, ‖fderiv ℝ (fderiv ℝ (fun y => f ((y,v),b))) x‖ ≤ D) (hf : MemLp f 2 (doubleMeasure Q)) (E : Set ℂ) (hE : MeasurableSet E) (hX : ∀ᵐ z ∂doubleMeasure Q, z.1.1 ∉ E → xDerivative f z = 0) (j : ℤ) : let hx := xDerivative_memLp_of_gradient_bound Q hm (fun v b => (hd v b).differentiable (by simp)) hC let hy := yDerivative_memLp_of_gradient_bound Q hm (fun v b => (hd v b).differentiable (by simp)) hC let R := restrictL2 (μ := doubleMeasure Q) {z : DoublePhase | z.1.1 ∈ E} (hE.preimage measurable_fst.fst) ∃ dx dy : Lp ℂ 2 (doubleMeasure Q), (geodesicHilbertFlow Q).HasGenerator ((angularCircleAction Q).projection j (hf.toLp _)) dx ∧ (transverseHilbertFlow Q).HasGenerator ((angularCircleAction Q).projection j (hf.toLp _)) dy ∧ ‖dx‖ ^ 2 ≤ 4 * ‖hx.toLp _‖ * ‖R (hy.toLp _)‖ ∧ ‖dy‖ ^ 2 ≤ 4 * ‖hx.toLp _‖ * ‖R (hy.toLp _)‖ := by dsimp only let hx := xDerivative_memLp_of_gradient_bound Q hm (fun v b => (hd v b).differentiable (by simp)) hC let hy := yDerivative_memLp_of_gradient_bound Q hm (fun v b => (hd v b).differentiable (by simp)) hC let R := restrictL2 (μ := doubleMeasure Q) {z : DoublePhase | z.1.1 ∈ E} (hE.preimage measurable_fst.fst) let a := (1/2 : ℂ) • (hx.toLp _ - Complex.I • hy.toLp _) let b := (1/2 : ℂ) • (hx.toLp _ + Complex.I • hy.toLp _) have hab : a + b = hx.toLp _ := by dsimp [a,b]; module have hsub : a - b = (-Complex.I) • hy.toLp _ := by dsimp [a,b]; module have hRx : R (hx.toLp _) = hx.toLp _ := by apply restrictL2_fixed filter_upwards [hx.coeFn_toLp,hX] with z hz hh simpa only [hz, mem_ofPred_eq] using hh have hn (k : ℤ) : ‖(angularCircleAction Q).projection (k+2) a‖ = ‖(angularCircleAction Q).projection k b‖ := by have hh := seam_projected_CR_norms_of_bounded_jets hs hm hd hC hD hf (k+1) have hi : k + 1 + 1 = k + 2 := by omega simpa only [hi, add_sub_cancel_right] using hh have hh := (angularCircleAction Q).localized_CR_energy a b j R (spatialRestriction_commute Q E hE) (restrictL2_selfadjoint _ _) (by rw [hab]; exact hRx) hn have hRsub : ‖R (a-b)‖ = ‖R (hy.toLp _)‖ := by rw [hsub, map_smul, norm_smul, norm_neg, Complex.norm_I, one_mul] rw [hab, hRsub] at hh have hcr := seam_rotationalCR hs (fun v b => (hd v b).differentiable (by simp)) hC hf hx hy have hg := hcr.projection j have heq : ‖(angularCircleAction Q).projection (j+1) a‖ = ‖(angularCircleAction Q).projection (j-1) b‖ := seam_projected_CR_norms_of_bounded_jets hs hm hd hC hD hf j let da := (angularCircleAction Q).projection (j+1) a let db := (angularCircleAction Q).projection (j-1) b refine ⟨da+db, Complex.I • (da-db), hg.1, hg.2, ?_, ?_⟩ · have hl : ‖da+db‖ ≤ 2 * ‖da‖ := by calc _ ≤ ‖da‖ + ‖db‖ := norm_add_le _ _ _ = _ := by rw [← heq]; ring calc _ ≤ (2*‖da‖)^2 := by gcongr _ = 4*‖da‖^2 := by ring _ ≤ _ := by linarith [hh] · have hl : ‖Complex.I • (da-db)‖ ≤ 2 * ‖da‖ := by rw [norm_smul, Complex.norm_I, one_mul] calc _ ≤ ‖da‖ + ‖db‖ := norm_sub_le _ _ _ = _ := by rw [← heq]; ring calc _ ≤ (2*‖da‖)^2 := by gcongr _ = 4*‖da‖^2 := by ring _ ≤ _ := by linarith [hh] end TriangularBilliards namespace TriangularBilliards open MeasureTheory Set namespace Triangle lemma clearance_sublevel (Q : Triangle) (R : ℝ) : {x | Q.clearance x ≤ R} = ⋃ i : Fin 3, Metric.closedBall (Q.vertex i) R := by ext x constructor · intro hx obtain ⟨y, ⟨i, rfl⟩, hi⟩ := (finite_range Q.vertex).isCompact.exists_infDist_eq_dist (range_nonempty Q.vertex) x exact mem_iUnion.mpr ⟨i, by simpa only [Metric.mem_closedBall, ← hi, clearance, mem_ofPred_eq] using hx⟩ · intro hx obtain ⟨i, hi⟩ := mem_iUnion.mp hx exact le_trans (Metric.infDist_le_dist_of_mem (mem_range_self i)) hi lemma clearance_sublevel_measurable (Q : Triangle) (R : ℝ) : MeasurableSet {x | Q.clearance x ≤ R} := measurableSet_le Q.clearance_lipschitz.continuous.measurable measurable_const lemma clearance_area_bound (Q : Triangle) (R : ℝ) : volume {x | Q.clearance x ≤ R} ≤ 3 * (ENNReal.ofReal R) ^ 2 * (NNReal.pi : ℝ≥0∞) := by rw [Q.clearance_sublevel] calc volume (⋃ i : Fin 3, Metric.closedBall (Q.vertex i) R) ≤ ∑' i : Fin 3, volume (Metric.closedBall (Q.vertex i) R) := measure_iUnion_le _ _ = 3 * (ENNReal.ofReal R) ^ 2 * (NNReal.pi : ℝ≥0∞) := by simp only [Complex.volume_closedBall, tsum_fintype, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] ring end Triangle /-- The small-tip neighborhood on the actual oriented double. -/ def unsafeSet (Q : Triangle) (R : ℝ) : Set DoublePhase := {z | Q.clearance z.1.1 ≤ R} lemma unsafeSet_measurable (Q : Triangle) (R : ℝ) : MeasurableSet (unsafeSet Q R) := (Q.clearance_sublevel_measurable R).preimage measurable_fst.fst lemma unsafeSet_measure_bound (Q : Triangle) (R : ℝ) : doubleMeasure Q (unsafeSet Q R) ≤ (volume Q.table)⁻¹ * (3 * (ENNReal.ofReal R) ^ 2 * (NNReal.pi : ℝ≥0∞)) := by have he : unsafeSet Q R = ({x : ℂ | Q.clearance x ≤ R} ×ˢ (univ : Set Circle)) ×ˢ (univ : Set (ZMod 2)) := by ext z; simp only [unsafeSet, mem_ofPred_eq, mem_prod, mem_univ, and_true] rw [he, doubleMeasure, Measure.prod_prod, measure_univ, mul_one, phaseMeasure, Measure.prod_prod, measure_univ, mul_one, Measure.smul_apply, smul_eq_mul] gcongr calc volume.restrict Q.table {x | Q.clearance x ≤ R} ≤ volume {x | Q.clearance x ≤ R} := Measure.restrict_apply_le _ _ _ ≤ _ := Q.clearance_area_bound R end TriangularBilliards namespace TriangularBilliards.Analysis open MeasureTheory Filter Set variable {A : Type uA} [MeasurableSpace A] {μ : Measure A} lemma restrictL2_norm_of_bounded (E : Set A) (hE : MeasurableSet E) (hμE : μ E ≠ ⊤) {u : Lp ℂ 2 μ} {H : ℝ} (hH : 0 ≤ H) (hu : ∀ᵐ z ∂μ, ‖u z‖ ≤ H) : ‖restrictL2 E hE u‖ ≤ H * (μ E).toReal ^ (1/2 : ℝ) := by calc _ ≤ ‖indicatorConstLp 2 hE hμE (H : ℂ)‖ := by apply Lp.norm_le_norm_of_ae_le filter_upwards [hu,restrictL2_coe E hE u, indicatorConstLp_coeFn (p := 2) (hs := hE) (hμs := hμE) (c := (H : ℂ))] with z hz he hc rw [he,hc] by_cases h : z ∈ E · simpa only [indicator_of_mem h, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hH] using hz · simp only [indicator_of_notMem h, le_refl] _ = _ := by rw [norm_indicatorConstLp (by norm_num : (2 : ℝ≥0∞) ≠ 0) (by norm_num : (2 : ℝ≥0∞) ≠ ⊤)] simp [Complex.norm_real, abs_of_nonneg hH, Measure.real] end TriangularBilliards.Analysis namespace TriangularBilliards open Analysis SpatialSmoothing MeasureTheory Filter Set open scoped Topology ContDiff NNReal local instance : Fact (0 < 2 * Real.pi) := ⟨by positivity⟩ lemma vertexCutoff_fderiv_zero_of_large (Q : Triangle) {R : ℝ} (hR : 0 < R) {x : ℂ} (hx : 2 * R < Q.clearance x) : fderiv ℝ (vertexCutoff Q R) x = 0 := by have he : vertexCutoff Q R =ᶠ[𝓝 x] fun _ => (1 : ℝ) := by filter_upwards [((isOpen_lt continuous_const Q.clearance_lipschitz.continuous).mem_nhds hx)] with y hy exact vertexCutoff_one Q hR hy.le rw [he.fderiv_eq, fderiv_const_apply] lemma supportedSmoothing_invariant_x_zero (Q : Triangle) {ε R A : ℝ} (hε : 0 < ε) (hR : Q.safetyFactor * ε < R) (hsmall : 2 * R * Q.coordinateBound < 1) (hA : 2 * R + ε ≤ A) {f : DoublePhase → ℂ} (hm : StronglyMeasurable f) (hf : MemLp f 2 (doubleMeasure Q)) (hfi : (geodesicHilbertFlow Q).HasGenerator (hf.toLp f) 0) {H : ℝ} (hH : ∀ z, ‖f z‖ ≤ H) : ∀ᵐ z ∂doubleMeasure Q, z ∉ unsafeSet Q A → xDerivative (supportedSmoothing Q ε R f) z = 0 := by have hRp : 0 < R := (mul_pos Q.safetyFactor_pos hε).trans hR filter_upwards [invariant_smoothing_direction_zero Q hε hR hsmall hA hm hf hfi hH] with z hz hza have hb : A < Q.clearance z.1.1 := lt_of_not_ge hza have hc : 2 * R < Q.clearance z.1.1 := by linarith rw [supportedSmoothing_xDerivative Q hε R hm hH, supportedSmoothingGradient, vertexCutoff_fderiv_zero_of_large Q hRp hc, vertexCutoff_one Q hRp hc.le, one_smul, ContinuousLinearMap.zero_smulRight, add_zero] exact hz hb.le lemma unsafeSet_real_measure_bound (Q : Triangle) {R : ℝ} (hR : 0 ≤ R) : (doubleMeasure Q (unsafeSet Q R)).toReal ≤ ((volume Q.table).toReal)⁻¹ * (3 * R ^ 2 * Real.pi) := by have ht : (volume Q.table)⁻¹ * (3 * (ENNReal.ofReal R)^2 * (NNReal.pi : ℝ≥0∞)) ≠ ⊤ := by exact ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr Q.area_pos.ne') (by finiteness) have hh := ENNReal.toReal_mono ht (unsafeSet_measure_bound Q R) simpa only [ENNReal.toReal_mul, ENNReal.toReal_inv, ENNReal.toReal_pow, ENNReal.toReal_ofNat, ENNReal.toReal_ofReal hR, ENNReal.coe_toReal, NNReal.coe_real_pi] using hh /-- Generic localized gradient control on an actual smooth seam field, with its exact tip-neighborhood measure retained. -/ lemma seam_tip_energy {Q : Triangle} {f : DoublePhase → ℂ} (hs : SeamCompatible Q f) (hm : Measurable f) (hd : ∀ v b, ContDiff ℝ ∞ (fun x => f ((x,v),b))) {C D : ℝ≥0} (hC : ∀ x v b, ‖fderiv ℝ (fun y => f ((y,v),b)) x‖ ≤ C) (hD : ∀ x v b, ‖fderiv ℝ (fderiv ℝ (fun y => f ((y,v),b))) x‖ ≤ D) (hf : MemLp f 2 (doubleMeasure Q)) (A : ℝ) (hX : ∀ᵐ z ∂doubleMeasure Q, z ∉ unsafeSet Q A → xDerivative f z = 0) (j : ℤ) : ∃ dx dy : Lp ℂ 2 (doubleMeasure Q), (geodesicHilbertFlow Q).HasGenerator ((angularCircleAction Q).projection j (hf.toLp _)) dx ∧ (transverseHilbertFlow Q).HasGenerator ((angularCircleAction Q).projection j (hf.toLp _)) dy ∧ ‖dx‖ ^ 2 ≤ 4 * (C : ℝ)^2 * (doubleMeasure Q (unsafeSet Q A)).toReal ∧ ‖dy‖ ^ 2 ≤ 4 * (C : ℝ)^2 * (doubleMeasure Q (unsafeSet Q A)).toReal := by let hx := xDerivative_memLp_of_gradient_bound Q hm (fun v b => (hd v b).differentiable (by simp)) hC let hy := yDerivative_memLp_of_gradient_bound Q hm (fun v b => (hd v b).differentiable (by simp)) hC let R := restrictL2 (μ := doubleMeasure Q) (unsafeSet Q A) (unsafeSet_measurable Q A) have hxb : ∀ᵐ z ∂doubleMeasure Q, ‖hx.toLp _ z‖ ≤ (C : ℝ) := by filter_upwards [hx.coeFn_toLp] with z hz rw [hz] exact (ContinuousLinearMap.le_opNorm _ _).trans (by rw [Circle.norm_coe, mul_one]; exact hC z.1.1 z.1.2 z.2) have hyb : ∀ᵐ z ∂doubleMeasure Q, ‖hy.toLp _ z‖ ≤ (C : ℝ) := by filter_upwards [hy.coeFn_toLp] with z hz rw [hz] exact (ContinuousLinearMap.le_opNorm _ _).trans (by rw [norm_transverseVelocity, mul_one]; exact hC z.1.1 z.1.2 z.2) have hrx : R (hx.toLp _) = hx.toLp _ := by apply restrictL2_fixed filter_upwards [hx.coeFn_toLp,hX] with z hz hh simpa only [hz] using hh have hxn : ‖hx.toLp _‖ ≤ (C : ℝ) * (doubleMeasure Q (unsafeSet Q A)).toReal ^ (1/2 : ℝ) := by rw [← hrx] exact restrictL2_norm_of_bounded _ _ (measure_ne_top _ _) C.coe_nonneg hxb have hyn := restrictL2_norm_of_bounded (unsafeSet Q A) (unsafeSet_measurable Q A) (measure_ne_top _ _) C.coe_nonneg hyb obtain ⟨dx,dy,hgx,hgy,hbx,hby⟩ := seam_localized_energy hs hm hd hC hD hf {x | Q.clearance x ≤ A} (Q.clearance_sublevel_measurable A) hX j have hmass : ((doubleMeasure Q (unsafeSet Q A)).toReal ^ (1/2 : ℝ))^2 = (doubleMeasure Q (unsafeSet Q A)).toReal := by rw [← Real.sqrt_eq_rpow, Real.sq_sqrt ENNReal.toReal_nonneg] have hb : 4 * ‖hx.toLp _‖ * ‖R (hy.toLp _)‖ ≤ 4 * (C : ℝ)^2 * (doubleMeasure Q (unsafeSet Q A)).toReal := by calc _ ≤ 4 * ((C : ℝ) * (doubleMeasure Q (unsafeSet Q A)).toReal ^ (1/2 : ℝ)) * ((C : ℝ) * (doubleMeasure Q (unsafeSet Q A)).toReal ^ (1/2 : ℝ)) := by gcongr _ = _ := by nlinarith [hmass] exact ⟨dx,dy,hgx,hgy,hbx.trans hb,hby.trans hb⟩ end TriangularBilliards end end end end end end end end end end OAI