| import OAI.Dynamics.TriangleBilliards.DerivativeAdjoints |
|
|
| 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 |
|
|
| namespace TriangularBilliards |
| open Analysis SpatialSmoothing Filter |
| open scoped NNReal ContDiff |
|
|
| lemma supportedSmoothing_xDerivative (Q : Triangle) {ε : ℝ} (hε : 0 < ε) (R : ℝ) |
| {f : DoublePhase → ℂ} (hm : StronglyMeasurable f) {H : ℝ} (hH : ∀ z, ‖f z‖ ≤ H) |
| (z : DoublePhase) : xDerivative (supportedSmoothing Q ε R f) z = |
| supportedSmoothingGradient Q ε R f z (z.1.2 : ℂ) := by |
| exact congrArg (fun L : ℂ →L[ℝ] ℂ => L (z.1.2 : ℂ)) |
| (supportedSmoothing_hasFDerivAt Q hε R hm hH z.1.2 z.2 z.1.1).fderiv |
|
|
| lemma supportedSmoothing_fderiv_bound (Q : Triangle) {ε R : ℝ} |
| (hε : 0 < ε) (hR : 0 < R) {f : DoublePhase → ℂ} (hm : StronglyMeasurable f) |
| {H : ℝ} (hf : ∀ z, ‖f z‖ ≤ H) : |
| ∃ C : ℝ≥0, ∀ x v b, ‖fderiv ℝ (fun y => supportedSmoothing Q ε R f ((y,v),b)) x‖ ≤ C := by |
| obtain ⟨D,hD,hDb⟩ := exists_vertexCutoff_derivative_bound |
| have hH : 0 ≤ H := (norm_nonneg _).trans (hf ((0,1),0)) |
| let C := 4 * derivativeMass * H / ε + (D / R) * (4 * H) |
| have hC : 0 ≤ C := by |
| have := derivativeMass_nonneg |
| dsimp [C] |
| positivity |
| refine ⟨⟨C,hC⟩, fun x v b => ?_⟩ |
| rw [(supportedSmoothing_hasFDerivAt Q hε R hm hf v b x).fderiv] |
| exact supportedSmoothingGradient_norm_bound Q hε hH hf (hDb Q R hR) ((x,v),b) |
|
|
| lemma supportedSmoothing_memLp (Q : Triangle) {ε : ℝ} (hε : 0 < ε) (R : ℝ) |
| {f : DoublePhase → ℂ} (hm : StronglyMeasurable f) {H : ℝ} (hf : ∀ z, ‖f z‖ ≤ H) : |
| MemLp (supportedSmoothing Q ε R f) 2 (doubleMeasure Q) := by |
| apply MemLp.of_bound (supportedSmoothing_stronglyMeasurable Q ε R hm).aestronglyMeasurable (4*H) |
| exact Filter.Eventually.of_forall (supportedSmoothing_norm_bound Q hε R |
| ((norm_nonneg _).trans (hf ((0,1),0))) hf) |
|
|
| lemma supportedSmoothing_xDerivative_memLp (Q : Triangle) {ε R : ℝ} |
| (hε : 0 < ε) (hR : 0 < R) {f : DoublePhase → ℂ} (hm : StronglyMeasurable f) |
| {H : ℝ} (hf : ∀ z, ‖f z‖ ≤ H) : |
| MemLp (xDerivative (supportedSmoothing Q ε R f)) 2 (doubleMeasure Q) := by |
| have he : xDerivative (supportedSmoothing Q ε R f) = |
| fun z => supportedSmoothingGradient Q ε R f z (z.1.2 : ℂ) := |
| funext (supportedSmoothing_xDerivative Q hε R hm hf) |
| obtain ⟨C,hC⟩ := supportedSmoothing_fderiv_bound Q hε hR hm hf |
| refine MemLp.of_bound ?_ (C : ℝ) ?_ |
| · rw [he] |
| exact (isBoundedBilinearMap_apply.continuous.comp_stronglyMeasurable |
| ((supportedSmoothingGradient_stronglyMeasurable Q ε R hm).prodMk |
| ((show Continuous (fun z : DoublePhase => (z.1.2 : ℂ)) from |
| continuous_subtype_val.comp (continuous_snd.comp continuous_fst)).stronglyMeasurable))).aestronglyMeasurable |
| · filter_upwards [] with z |
| exact (ContinuousLinearMap.le_opNorm _ _).trans (by |
| rw [Circle.norm_coe, mul_one] |
| exact hC z.1.1 z.1.2 z.2) |
|
|
| lemma supportedSmoothing_generator (Q : Triangle) {ε R : ℝ} |
| (hε : 0 < ε) (hR : Q.safetyFactor * ε < R) |
| (hsmall : 2 * R * Q.coordinateBound < 1) |
| {f : DoublePhase → ℂ} (hm : StronglyMeasurable f) {H : ℝ} (hf : ∀ z, ‖f z‖ ≤ H) : |
| let hs := supportedSmoothing_memLp Q hε R hm hf |
| let hx := supportedSmoothing_xDerivative_memLp Q hε |
| ((mul_pos Q.safetyFactor_pos hε).trans hR) hm hf |
| (geodesicHilbertFlow Q).HasGenerator (hs.toLp _) (hx.toLp _) := by |
| obtain ⟨C,hC⟩ := supportedSmoothing_fderiv_bound Q hε |
| ((mul_pos Q.safetyFactor_pos hε).trans hR) hm hf |
| apply seam_geodesic_generator (supportedSmoothing_seam Q hε hR hsmall f) |
| (fun v b => (supportedSmoothing_contDiff Q hε R hm hf v b).differentiable (by simp)) hC |
|
|
| end TriangularBilliards |
|
|
| namespace TriangularBilliards |
| open Analysis SpatialSmoothing Filter |
| open scoped Topology ComplexConjugate |
|
|
| lemma supportedSmoothing_eq_of_supported (Q : Triangle) {ε R A : ℝ} |
| (hε : 0 < ε) (hR : 0 < R) (hA : 2 * R + ε ≤ A) |
| {f : DoublePhase → ℂ} (hf : SpatiallyZero Q A f) {z : DoublePhase} |
| (hz : z.1.1 ∈ Q.table) : |
| supportedSmoothing Q ε R f z = reflectedSmoothing Q ε f z := by |
| by_cases hr : 2 * R ≤ Q.clearance z.1.1 |
| · simp only [supportedSmoothing, vertexCutoff_one Q hR hr, one_smul] |
| · have hzero := reflectedSmoothing_zero_near_tips Q hε hf hz (by linarith) |
| simp only [supportedSmoothing, hzero, smul_zero] |
|
|
| lemma supportedSmoothing_derivative_eq_of_supported (Q : Triangle) {ε R A : ℝ} |
| (hε : 0 < ε) (hR : 0 < R) (hA : 2 * R + ε ≤ A) |
| {f : DoublePhase → ℂ} (hf : SpatiallyZero Q A f) (hm : StronglyMeasurable f) |
| {H : ℝ} (hH : ∀ z, ‖f z‖ ≤ H) {z : DoublePhase} (hz : z.1.1 ∈ Q.table) : |
| xDerivative (supportedSmoothing Q ε R f) z = reflectedSmoothingGradient Q ε f z (z.1.2 : ℂ) := by |
| have he : (fun y => supportedSmoothing Q ε R f ((y,z.1.2),z.2)) =ᶠ[𝓝 z.1.1] |
| (fun y => reflectedSmoothing Q ε f ((y,z.1.2),z.2)) := by |
| filter_upwards [isOpen_interior.eventually_mem hz] with y hy |
| exact supportedSmoothing_eq_of_supported Q hε hR hA hf hy |
| change fderiv ℝ (fun y => supportedSmoothing Q ε R f ((y,z.1.2),z.2)) z.1.1 _ = _ |
| rw [he.fderiv_eq] |
| exact congrArg (fun L : ℂ →L[ℝ] ℂ => L (z.1.2 : ℂ)) |
| (reflectedSmoothing_hasFDerivAt Q hε hm (bounded_spatialSlice_integrable Q hm hH z.1.2 z.2) |
| (fun i => bounded_spatialSlice_integrable Q hm hH (reflectedDirection Q i z.1.2) (z.2+1)) z.1.1).fderiv |
|
|
| lemma reflectedSmoothing_direction_memLp (Q : Triangle) {ε : ℝ} (hε : 0 < ε) |
| {f : DoublePhase → ℂ} (hm : StronglyMeasurable f) (hf : MemLp f 2 (doubleMeasure Q)) : |
| MemLp (fun z => reflectedSmoothingGradient Q ε f z (z.1.2 : ℂ)) 2 (doubleMeasure Q) := |
| gradientApply_memLp (reflectedSmoothingGradient_memLp Q hε hm hf) |
| ((show Continuous (fun z : DoublePhase => (z.1.2 : ℂ)) from |
| continuous_subtype_val.comp (continuous_snd.comp continuous_fst)).stronglyMeasurable) |
| (fun _z => (Circle.norm_coe _).le) |
|
|
| lemma reflectedSmoothing_supported_generator (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) {H : ℝ} (hH : ∀ z, ‖f z‖ ≤ H) |
| (hf : MemLp f 2 (doubleMeasure Q)) (hs : SpatiallyZero Q A f) : |
| (geodesicHilbertFlow Q).HasGenerator |
| ((reflectedSmoothing_memLp Q hε hm hf).toLp _) |
| ((reflectedSmoothing_direction_memLp Q hε hm hf).toLp _) := by |
| have hRp := (mul_pos Q.safetyFactor_pos hε).trans hR |
| have h := supportedSmoothing_generator Q hε hR hsmall hm hH |
| have h₁ := (supportedSmoothing_memLp Q hε R hm hH).toLp_congr |
| (reflectedSmoothing_memLp Q hε hm hf) (by |
| filter_upwards [ae_double_position_in_table Q] with z hz |
| exact supportedSmoothing_eq_of_supported Q hε hRp hA hs hz) |
| have h₂ := (supportedSmoothing_xDerivative_memLp Q hε hRp hm hH).toLp_congr |
| (reflectedSmoothing_direction_memLp Q hε hm hf) (by |
| filter_upwards [ae_double_position_in_table Q] with z hz |
| exact supportedSmoothing_derivative_eq_of_supported Q hε hRp hA hs hm hH hz) |
| change (geodesicHilbertFlow Q).HasGenerator |
| ((supportedSmoothing_memLp Q hε R hm hH).toLp _) _ at h |
| rwa [h₁, h₂] at h |
|
|
| lemma integral_pairing_toLp {A : Type uA} [MeasurableSpace A] {μ : Measure A} |
| {f g : A → ℂ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) : |
| (∫ z, f z * conj (g z) ∂μ) = inner ℂ (hg.toLp g) (hf.toLp f) := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hf.coeFn_toLp, hg.coeFn_toLp] with z hz hzz |
| simp only [hz, hzz, RCLike.inner_apply] |
|
|
| /-- Exact local kernel-transport commutation for a bounded invariant field, |
| expressed against arbitrary bounded fields with a sufficient vertex margin. -/ |
| lemma invariant_smoothing_direction_pairing_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 g : DoublePhase → ℂ} (hfm : StronglyMeasurable f) (hgm : StronglyMeasurable g) |
| (hf : MemLp f 2 (doubleMeasure Q)) (hg : MemLp g 2 (doubleMeasure Q)) |
| (hfi : (geodesicHilbertFlow Q).HasGenerator (hf.toLp f) 0) |
| {H : ℝ} (hH : ∀ z, ‖g z‖ ≤ H) (hgs : SpatiallyZero Q A g) : |
| (∫ z, reflectedSmoothingGradient Q ε f z (z.1.2 : ℂ) * conj (g z) ∂doubleMeasure Q) = 0 := by |
| have h := reflectedSmoothingGradient_adjoint Q hε |
| (V := fun c => (c.1 : ℂ)) (show Continuous (fun c : DirectionParity => (c.1 : ℂ)) from |
| continuous_subtype_val.comp continuous_fst).stronglyMeasurable |
| (fun c => (Circle.norm_coe _).le) (fun i c => rfl) hfm hgm hf hg |
| have hgen := reflectedSmoothing_supported_generator Q hε hR hsmall hA hgm hH hg hgs |
| have hsk := (geodesicHilbertFlow Q).generator_skew hgen hfi |
| simp only [inner_zero_right, zero_add] at hsk |
| rw [h, integral_pairing_toLp hf (reflectedSmoothing_direction_memLp Q hε hgm hg), hsk, neg_zero] |
|
|
| end TriangularBilliards |
|
|
| namespace TriangularBilliards |
| open Analysis SpatialSmoothing Filter |
| open scoped Topology ComplexConjugate |
|
|
| lemma reflectedSmoothing_direction_norm_bound (Q : Triangle) {ε : ℝ} (hε : 0 < ε) |
| {f : DoublePhase → ℂ} {H : ℝ} (hH : 0 ≤ H) (hf : ∀ z, ‖f z‖ ≤ H) |
| (z : DoublePhase) : ‖reflectedSmoothingGradient Q ε f z (z.1.2 : ℂ)‖ ≤ |
| 4 * derivativeMass * H / ε := by |
| calc |
| _ ≤ ‖reflectedSmoothingGradient Q ε f z‖ * ‖(z.1.2 : ℂ)‖ := ContinuousLinearMap.le_opNorm _ _ |
| _ ≤ _ := by rw [Circle.norm_coe, mul_one]; exact reflectedSmoothingGradient_norm_bound Q hε hH hf z |
|
|
| lemma reflectedSmoothing_direction_stronglyMeasurable (Q : Triangle) (ε : ℝ) |
| {f : DoublePhase → ℂ} (hm : StronglyMeasurable f) : |
| StronglyMeasurable (fun z => reflectedSmoothingGradient Q ε f z (z.1.2 : ℂ)) := |
| isBoundedBilinearMap_apply.continuous.comp_stronglyMeasurable |
| ((reflectedSmoothingGradient_stronglyMeasurable Q ε hm).prodMk |
| (show Continuous (fun z : DoublePhase => (z.1.2 : ℂ)) from |
| continuous_subtype_val.comp (continuous_snd.comp continuous_fst)).stronglyMeasurable) |
|
|
| /-- The actual local X-commutation equation for a bounded invariant field. |
| No local distributional transport statement is used as an extra hypothesis. -/ |
| lemma invariant_smoothing_direction_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} (hfm : 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, A ≤ Q.clearance z.1.1 → |
| reflectedSmoothingGradient Q ε f z (z.1.2 : ℂ) = 0 := by |
| let D : DoublePhase → ℂ := fun z => reflectedSmoothingGradient Q ε f z (z.1.2 : ℂ) |
| let E : Set DoublePhase := {z | A ≤ Q.clearance z.1.1} |
| let g : DoublePhase → ℂ := E.indicator D |
| have hE : MeasurableSet E := |
| measurableSet_le measurable_const |
| (Q.clearance_lipschitz.continuous.measurable.comp (measurable_fst.fst)) |
| have hDm : StronglyMeasurable D := reflectedSmoothing_direction_stronglyMeasurable Q ε hfm |
| have hDp : MemLp D 2 (doubleMeasure Q) := reflectedSmoothing_direction_memLp Q hε hfm hf |
| have hgm : StronglyMeasurable g := hDm.indicator hE |
| have hgp : MemLp g 2 (doubleMeasure Q) := hDp.indicator hE |
| have hH₀ : 0 ≤ H := (norm_nonneg _).trans (hH ((0,1),0)) |
| have hgb : ∀ z, ‖g z‖ ≤ 4 * derivativeMass * H / ε := by |
| intro z |
| by_cases hz : z ∈ E |
| · change ‖E.indicator D z‖ ≤ _ |
| rw [Set.indicator_of_mem hz] |
| exact reflectedSmoothing_direction_norm_bound Q hε hH₀ hH z |
| · change ‖E.indicator D z‖ ≤ _ |
| rw [Set.indicator_of_notMem hz, norm_zero] |
| exact div_nonneg (mul_nonneg (mul_nonneg (by norm_num) derivativeMass_nonneg) hH₀) hε.le |
| have hgs : SpatiallyZero Q A g := by |
| intro y _ hy v b |
| exact Set.indicator_of_notMem (show ((y,v),b) ∉ E from not_le.mpr hy) D |
| have hp := invariant_smoothing_direction_pairing_zero Q hε hR hsmall hA hfm hgm hf hgp hfi hgb hgs |
| have he : (∫ z, g z * conj (g z) ∂doubleMeasure Q) = 0 := by |
| rw [← hp] |
| apply integral_congr_ae |
| filter_upwards [] with z |
| by_cases hz : z ∈ E |
| · simp only [g, Set.indicator_of_mem hz, D] |
| · simp only [g, Set.indicator_of_notMem hz, map_zero, mul_zero] |
| rw [integral_pairing_toLp hgp hgp, inner_self_eq_zero] at he |
| have hgz : g =ᵐ[doubleMeasure Q] 0 := by |
| have hh := hgp.coeFn_toLp |
| rw [he] at hh |
| exact hh.symm.trans (Lp.coeFn_zero _ _ _) |
| filter_upwards [hgz] with z hz hza |
| change D z = 0 |
| simpa only [g, Set.indicator_of_mem (show z ∈ E from hza), Pi.zero_apply] using hz |
|
|
| end TriangularBilliards |
|
|
| namespace TriangularBilliards.Analysis |
| open Filter |
| open scoped ComplexConjugate |
|
|
| lemma pairing_add_sum_symmetric {A : Type uA} [MeasurableSpace A] {μ : Measure A} |
| {f g d e : A → ℂ} {r s : Fin 3 → A → ℂ} |
| (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) |
| (hd : MemLp d 2 μ) (he : MemLp e 2 μ) |
| (hr : ∀ i, MemLp (r i) 2 μ) (hs : ∀ i, MemLp (s i) 2 μ) |
| (h0 : (∫ z, d z * conj (g z) ∂μ) = (∫ z, f z * conj (e z) ∂μ)) |
| (hi : ∀ i, (∫ z, r i z * conj (g z) ∂μ) = (∫ z, f z * conj (s i z) ∂μ)) : |
| (∫ z, (d z + ∑ i, r i z) * conj (g z) ∂μ) = |
| (∫ z, f z * conj (e z + ∑ i, s i z) ∂μ) := by |
| have hD : Integrable (fun z => d z * conj (g z)) μ := hd.integrable_mul (memLp_conj hg) |
| have hE : Integrable (fun z => f z * conj (e z)) μ := hf.integrable_mul (memLp_conj he) |
| have hR (i) : Integrable (fun z => r i z * conj (g z)) μ := (hr i).integrable_mul (memLp_conj hg) |
| have hS (i) : Integrable (fun z => f z * conj (s i z)) μ := hf.integrable_mul (memLp_conj (hs i)) |
| simp only [add_mul, Finset.sum_mul, map_add, map_sum, mul_add, Finset.mul_sum] |
| rw [integral_add hD (integrable_finsetSum Finset.univ (fun i _ => hR i)), |
| integral_add hE (integrable_finsetSum Finset.univ (fun i _ => hS i)), |
| integral_finsetSum Finset.univ (fun i _ => hR i), |
| integral_finsetSum Finset.univ (fun i _ => hS i), h0] |
| simp only [hi] |
|
|
| end TriangularBilliards.Analysis |
|
|
| namespace TriangularBilliards |
| open Analysis SpatialSmoothing Filter |
| open scoped Topology ComplexConjugate |
|
|
| lemma spatialKernel_stronglyMeasurable (ε : ℝ) {R : ℂ → ℂ} (hR : Measurable R) : |
| StronglyMeasurable (fun p : (ℂ × DirectionParity) × ℂ => kernel ε (p.1.1-R p.2)) := |
| ((kernel_contDiff ε).continuous.measurable.comp |
| (measurable_fst.fst.sub (hR.comp measurable_snd))).stronglyMeasurable |
|
|
| lemma directS_adjoint (Q : Triangle) {ε : ℝ} (hε : 0 < ε) |
| {f g : DoublePhase → ℂ} (hfm : StronglyMeasurable f) (hgm : StronglyMeasurable g) |
| (hf : MemLp f 2 (doubleMeasure Q)) (hg : MemLp g 2 (doubleMeasure Q)) : |
| (∫ z, directS Q ε f z.2 z.1.2 z.1.1 * conj (g z) ∂doubleMeasure Q) = |
| (∫ z, f z * conj (directS Q ε g z.2 z.1.2 z.1.1) ∂doubleMeasure Q) := by |
| obtain ⟨K,hK⟩ := (kernel_hasCompactSupport hε).exists_bound_of_continuous (kernel_contDiff ε).continuous |
| exact doubleKernel_adjoint Q (J := id) (MeasurePreserving.id _) MeasurableEmbedding.id (fun _ => rfl) |
| (spatialKernel_stronglyMeasurable ε measurable_id) (spatialKernel_stronglyMeasurable ε measurable_id) |
| (fun x y _ => by simpa only [Real.norm_eq_abs] using hK (x-y)) |
| (fun x y _ => by simpa only [Real.norm_eq_abs] using hK (x-y)) |
| (fun x y _ => kernel_swap ε x y) hfm hgm hf hg |
|
|
| lemma reflectedS_adjoint (Q : Triangle) (i : Fin 3) {ε : ℝ} (hε : 0 < ε) |
| {f g : DoublePhase → ℂ} (hfm : StronglyMeasurable f) (hgm : StronglyMeasurable g) |
| (hf : MemLp f 2 (doubleMeasure Q)) (hg : MemLp g 2 (doubleMeasure Q)) : |
| (∫ z, reflectedS Q i ε f z.2 z.1.2 z.1.1 * conj (g z) ∂doubleMeasure Q) = |
| (∫ z, f z * conj (reflectedS Q i ε g z.2 z.1.2 z.1.1) ∂doubleMeasure Q) := by |
| obtain ⟨K,hK⟩ := (kernel_hasCompactSupport hε).exists_bound_of_continuous (kernel_contDiff ε).continuous |
| exact doubleKernel_adjoint Q (measurePreserving_directionParity Q i) |
| (measurableEmbedding_directionParity Q i) (directionParityReflection_involutive Q i) |
| (spatialKernel_stronglyMeasurable ε (wallReflection_contDiff Q i).continuous.measurable) |
| (spatialKernel_stronglyMeasurable ε (wallReflection_contDiff Q i).continuous.measurable) |
| (fun x y _ => by simpa only [Real.norm_eq_abs] using hK (x-wallReflection Q i y)) |
| (fun x y _ => by simpa only [Real.norm_eq_abs] using hK (x-wallReflection Q i y)) |
| (fun x y _ => kernel_wall_swap Q i ε x y) hfm hgm hf hg |
|
|
| lemma directS_memLp (Q : Triangle) {ε : ℝ} (hε : 0 < ε) |
| {f : DoublePhase → ℂ} (hfm : StronglyMeasurable f) (hf : MemLp f 2 (doubleMeasure Q)) : |
| MemLp (fun z => directS Q ε f z.2 z.1.2 z.1.1) 2 (doubleMeasure Q) := |
| (directS_eLpNorm Q hε hfm).trans_lt hf.eLpNorm_lt_top |
|
|
| lemma reflectedS_memLp (Q : Triangle) (i : Fin 3) {ε : ℝ} (hε : 0 < ε) |
| {f : DoublePhase → ℂ} (hfm : StronglyMeasurable f) (hf : MemLp f 2 (doubleMeasure Q)) : |
| MemLp (fun z => reflectedS Q i ε f z.2 z.1.2 z.1.1) 2 (doubleMeasure Q) := |
| (reflectedS_eLpNorm Q i hε hfm).trans_lt hf.eLpNorm_lt_top |
|
|
| lemma reflectedSmoothing_adjoint (Q : Triangle) {ε : ℝ} (hε : 0 < ε) |
| {f g : DoublePhase → ℂ} (hfm : StronglyMeasurable f) (hgm : StronglyMeasurable g) |
| (hf : MemLp f 2 (doubleMeasure Q)) (hg : MemLp g 2 (doubleMeasure Q)) : |
| (∫ z, reflectedSmoothing Q ε f z * conj (g z) ∂doubleMeasure Q) = |
| (∫ z, f z * conj (reflectedSmoothing Q ε g z) ∂doubleMeasure Q) := by |
| exact pairing_add_sum_symmetric hf hg (directS_memLp Q hε hfm hf) (directS_memLp Q hε hgm hg) |
| (fun i => reflectedS_memLp Q i hε hfm hf) (fun i => reflectedS_memLp Q i hε hgm hg) |
| (directS_adjoint Q hε hfm hgm hf hg) (fun i => reflectedS_adjoint Q i hε hfm hgm hf hg) |
|
|
| end TriangularBilliards |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
