import OAI.Dynamics.TriangleBilliards.Localization namespace OAI 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 open Analysis SpatialSmoothing MeasureTheory Filter Set open scoped Topology ContDiff NNReal local instance : Fact (0 < 2 * Real.pi) := ⟨by positivity⟩ /-- The uniform per-coefficient estimate for cutoff regularizations of an actual bounded invariant. The constant depends on the triangle, not the Fourier index, smoothing scale, or the bounded invariant. -/ lemma bounded_invariant_regularized_modes (Q : Triangle) : ∃ C : ℝ, 0 ≤ C ∧ ∀ {H : ℝ} {f : DoublePhase → ℂ} (hm : StronglyMeasurable f) (hf : MemLp f 2 (doubleMeasure Q)) (hH : ∀ z, ‖f z‖ ≤ H) (_hfi : (geodesicHilbertFlow Q).HasGenerator (hf.toLp f) 0) {ε : ℝ} (hε : 0 < ε) (_hsmall : 2 * ((Q.safetyFactor + 1) * ε) * Q.coordinateBound < 1) (j : ℤ), let w := supportedSmoothing Q ε ((Q.safetyFactor + 1) * ε) f let hw := supportedSmoothing_memLp Q hε ((Q.safetyFactor + 1) * ε) hm hH ∃ dx dy : Lp ℂ 2 (doubleMeasure Q), (geodesicHilbertFlow Q).HasGenerator ((angularCircleAction Q).projection j (hw.toLp w)) dx ∧ (transverseHilbertFlow Q).HasGenerator ((angularCircleAction Q).projection j (hw.toLp w)) dy ∧ ‖dx‖ ^ 2 ≤ C * H^2 ∧ ‖dy‖ ^ 2 ≤ C * H^2 := by obtain ⟨D,hDp,hD⟩ := exists_vertexCutoff_derivative_bound let K := Q.safetyFactor + 1 have hK : 1 < K := by dsimp [K]; linarith [Q.safetyFactor_pos] let J := 4 * derivativeMass + 4 * D have hJ : 0 ≤ J := by dsimp [J] exact add_nonneg (mul_nonneg (by norm_num) derivativeMass_nonneg) (mul_nonneg (by norm_num) hDp) let B := ((volume Q.table).toReal)⁻¹ * (3 * (2*K+1)^2 * Real.pi) have hB : 0 ≤ B := by dsimp [B]; positivity refine ⟨4 * J^2 * B, by positivity, ?_⟩ intro H f hm hf hH hfi ε hε hsmall j have hH₀ : 0 ≤ H := (norm_nonneg _).trans (hH ((0,1),0)) have hR : Q.safetyFactor * ε < K * ε := by dsimp [K]; nlinarith have hRp : 0 < K * ε := mul_pos (by linarith) hε let Cε : ℝ≥0 := ⟨J*H/ε, by positivity⟩ have hC (x : ℂ) (v : Circle) (b : ZMod 2) : ‖fderiv ℝ (fun y => supportedSmoothing Q ε (K*ε) f ((y,v),b)) x‖ ≤ Cε := by rw [(supportedSmoothing_hasFDerivAt Q hε (K*ε) hm hH v b x).fderiv] apply (supportedSmoothingGradient_norm_bound Q hε hH₀ hH (hD Q (K*ε) hRp) ((x,v),b)).trans change 4*derivativeMass*H/ε + (D/(K*ε))*(4*H) ≤ J*H/ε have hh : D / (K*ε) ≤ D / ε := div_le_div_of_nonneg_left hDp hε (by nlinarith) calc _ ≤ 4*derivativeMass*H/ε + (D/ε)*(4*H) := by gcongr _ = _ := by dsimp [J]; ring obtain ⟨M,hM⟩ := supportedSmoothing_second_bound Q hε hRp hm hH have hm' := (supportedSmoothing_stronglyMeasurable Q ε (K*ε) hm).measurable have hw := supportedSmoothing_memLp Q hε (K*ε) hm hH have hs := supportedSmoothing_seam Q hε hR hsmall f have hd := supportedSmoothing_contDiff Q hε (K*ε) hm hH have hA : 2 * (K*ε) + ε ≤ (2*K+1)*ε := by linarith obtain ⟨dx,dy,hgx,hgy,hbx,hby⟩ := seam_tip_energy hs hm' hd hC hM hw ((2*K+1)*ε) (supportedSmoothing_invariant_x_zero Q hε hR hsmall hA hm hf hfi hH) j have hmass : (doubleMeasure Q (unsafeSet Q ((2*K+1)*ε))).toReal ≤ B * ε^2 := by apply (unsafeSet_real_measure_bound Q (show 0 ≤ (2*K+1)*ε by positivity)).trans_eq dsimp [B] ring have hb : 4 * (Cε : ℝ)^2 * (doubleMeasure Q (unsafeSet Q ((2*K+1)*ε))).toReal ≤ (4 * J^2 * B) * H^2 := by calc _ ≤ 4 * (Cε : ℝ)^2 * (B*ε^2) := by gcongr _ = _ := by change 4 * (J*H/ε)^2 * (B*ε^2) = _ field_simp exact ⟨dx,dy,hgx,hgy,hbx.trans hb,hby.trans hb⟩ end TriangularBilliards namespace TriangularBilliards open Analysis SpatialSmoothing MeasureTheory Filter Set open scoped Topology ContDiff NNReal BoundedContinuousFunction lemma kernel_eq_zero_of_large {ε : ℝ} (hε : 0 < ε) {y : ℂ} (hy : ε ≤ ‖y‖) : kernel ε y = 0 := by by_contra h have hx : y ∈ support (kernel ε) := h rw [kernel_support hε, mem_ball_zero_iff] at hx exact (not_lt_of_ge hy) hx lemma fullS_tendsto (f : ℂ →ᵇ ℂ) (x : ℂ) : Tendsto (fun ε : ℝ => ∫ y : ℂ, kernel ε (x-y) • f y) (𝓝[>] 0) (𝓝 (f x)) := by have hs : Tendsto (fun ε : ℝ => support (kernel ε)) (𝓝[>] 0) (𝓝 (0 : ℂ)).smallSets := by rw [tendsto_smallSets_iff] intro s hs obtain ⟨r,hr,hb⟩ := Metric.mem_nhds_iff.mp hs filter_upwards [self_mem_nhdsWithin, nhdsWithin_le_nhds (gt_mem_nhds hr)] with ε hp he rw [kernel_support hp] exact (Metric.ball_subset_ball he.le).trans hb have hh := convolution_tendsto_right (μ := (volume : Measure ℂ)) (g := fun _ : ℝ => (f : ℂ → ℂ)) (k := fun _ : ℝ => x) (Eventually.of_forall (fun ε => kernel_nonneg ε)) (show ∀ᶠ ε in 𝓝[>] (0 : ℝ), ∫ y, kernel ε y = 1 from (show ∀ᶠ ε in 𝓝[>] (0 : ℝ), 0 < ε from self_mem_nhdsWithin).mono (fun ε hε => kernel_integral hε)) hs (Eventually.of_forall fun _ => f.continuous.aestronglyMeasurable) (f.continuous.continuousAt.tendsto.comp tendsto_snd) tendsto_const_nhds apply hh.congr (fun ε => ?_) rw [MeasureTheory.convolution_def] rw [← integral_sub_left_eq_self (fun y : ℂ => kernel ε (x-y) • f y) volume x] simp only [sub_sub_cancel, ContinuousLinearMap.lsmul_apply] lemma directS_eq_full (Q : Triangle) {ε : ℝ} (hε : 0 < ε) (f : DoublePhase → ℂ) (z : DoublePhase) (hz : closedBall z.1.1 ε ⊆ Q.table) : directS Q ε f z.2 z.1.2 z.1.1 = ∫ y : ℂ, kernel ε (z.1.1-y) • f ((y,z.1.2),z.2) := by rw [directS, ← integral_indicator Q.measurableSet_table] apply integral_congr_ae exact Eventually.of_forall fun y => by by_cases hy : y ∈ Q.table · rw [indicator_of_mem hy] · rw [indicator_of_notMem hy] have hn : ε < ‖z.1.1-y‖ := by by_contra hh exact hy (hz (by rw [mem_closedBall, dist_comm, dist_eq_norm]; exact not_lt.mp hh)) simp only [kernel_eq_zero_of_large hε hn.le, zero_smul] lemma reflectedS_zero_of_coord_phase (Q : Triangle) (i : Fin 3) {ε : ℝ} (hε : 0 < ε) (f : DoublePhase → ℂ) (z : DoublePhase) (hz : Q.coordinateBound * ε < Q.basis.coord (i+2) z.1.1) : reflectedS Q i ε f z.2 z.1.2 z.1.1 = 0 := by apply integral_eq_zero_of_ae filter_upwards [ae_restrict_mem Q.measurableSet_table] with y hy have hn : ε < ‖z.1.1 - wallReflection Q i y‖ := by by_contra hh exact (not_le_of_gt hz) (Q.coord_small_of_reflected_near i hy (not_lt.mp hh)) simp only [kernel_eq_zero_of_large hε hn.le, zero_smul, Pi.zero_apply] lemma eventually_interior_smoothing (Q : Triangle) (f : DoublePhase → ℂ) (z : DoublePhase) (hz : z.1.1 ∈ Q.table) : ∀ᶠ ε in 𝓝[>] (0 : ℝ), reflectedSmoothing Q ε f z = ∫ y : ℂ, kernel ε (z.1.1-y) • f ((y,z.1.2),z.2) := by obtain ⟨r,hr,hball⟩ := Metric.isOpen_iff.mp Q.isOpen_table _ hz have hcoords : ∀ᶠ ε in 𝓝 (0 : ℝ), ∀ i : Fin 3, Q.coordinateBound * ε < Q.basis.coord (i+2) z.1.1 := by rw [Filter.eventually_all] intro i have hp := (Q.table_iff_coords z.1.1).mp hz (i+2) have hc : Continuous (fun ε : ℝ => Q.coordinateBound * ε) := by fun_prop simpa only [mul_zero] using hc.continuousAt.eventually (gt_mem_nhds (by simpa only [mul_zero] using hp)) filter_upwards [nhdsWithin_le_nhds hcoords, nhdsWithin_le_nhds (gt_mem_nhds hr), self_mem_nhdsWithin] with ε hc hεr hε have hfull := directS_eq_full Q hε f z (fun y hy => hball (by rw [mem_ball] exact (mem_closedBall.mp hy).trans_lt hεr)) change directS Q ε f z.2 z.1.2 z.1.1 + ∑ i, reflectedS Q i ε f z.2 z.1.2 z.1.1 = _ simp only [reflectedS_zero_of_coord_phase Q _ hε f z (hc _), Finset.sum_const_zero, add_zero] exact hfull lemma smoothing_pointwise (Q : Triangle) (f : DoublePhase →ᵇ ℂ) (z : DoublePhase) (hz : z.1.1 ∈ Q.table) : Tendsto (fun ε : ℝ => reflectedSmoothing Q ε f z) (𝓝[>] 0) (𝓝 (f z)) := by let g : ℂ →ᵇ ℂ := f.compContinuous ⟨fun y => ((y,z.1.2),z.2), by fun_prop⟩ exact (fullS_tendsto g z.1.1).congr' ((eventually_interior_smoothing Q f z hz).mono (fun _ h => h.symm)) end TriangularBilliards namespace TriangularBilliards open Analysis SpatialSmoothing MeasureTheory Filter Set open scoped Topology ContDiff NNReal BoundedContinuousFunction lemma Triangle.clearance_pos_of_mem_table (Q : Triangle) {x : ℂ} (hx : x ∈ Q.table) : 0 < Q.clearance x := by apply ((finite_range Q.vertex).isClosed.notMem_iff_infDist_pos (range_nonempty _)).mp rintro ⟨i,rfl⟩ have h := (Q.table_iff_coords _).mp hx (i+2) have hv : Q.basis.coord (i+2) (Q.vertex i) = 0 := Q.basis.coord_apply (i+2) i |>.trans (ite_eq_right ((by decide : ∀ i : Fin 3, i+2 ≠ i) i)) linarith lemma supported_smoothing_pointwise (Q : Triangle) (K : ℝ) (hK : 0 < K) (f : DoublePhase →ᵇ ℂ) (z : DoublePhase) (hz : z.1.1 ∈ Q.table) : Tendsto (fun ε : ℝ => supportedSmoothing Q ε (K*ε) f z) (𝓝[>] 0) (𝓝 (f z)) := by have hc : ∀ᶠ ε in 𝓝 (0 : ℝ), 2*(K*ε) < Q.clearance z.1.1 := by have h : Continuous (fun ε : ℝ => 2*(K*ε)) := by fun_prop simpa only [mul_zero] using h.continuousAt.eventually (gt_mem_nhds (by simpa only [mul_zero] using Q.clearance_pos_of_mem_table hz)) apply (smoothing_pointwise Q f z hz).congr' filter_upwards [self_mem_nhdsWithin, nhdsWithin_le_nhds hc] with ε hε hh simp only [supportedSmoothing, vertexCutoff_one Q (mul_pos hK hε) hh.le, one_smul] lemma supported_smoothing_lpNorm_tendsto_bounded (Q : Triangle) {K : ℝ} (hK : 0 < K) (f : DoublePhase →ᵇ ℂ) : Tendsto (fun ε : ℝ => lpNorm (supportedSmoothing Q ε (K*ε) f - (f : DoublePhase → ℂ)) 2 (doubleMeasure Q)) (𝓝[>] 0) (𝓝 0) := by have hi : Tendsto (fun ε : ℝ => ∫ z, ‖supportedSmoothing Q ε (K*ε) f z - f z‖^2 ∂doubleMeasure Q) (𝓝[>] 0) (𝓝 (∫ _ : DoublePhase, (0 : ℝ) ∂doubleMeasure Q)) := by apply tendsto_integral_filter_of_norm_le_const · exact Eventually.of_forall fun ε => (((supportedSmoothing_stronglyMeasurable Q ε (K*ε) f.continuous.stronglyMeasurable).sub f.continuous.stronglyMeasurable).measurable.norm.pow_const 2).aestronglyMeasurable · refine ⟨(5*‖f‖)^2, ?_⟩ filter_upwards [self_mem_nhdsWithin] with ε hε exact Eventually.of_forall fun z => by rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] apply pow_le_pow_left₀ (norm_nonneg _) exact (norm_sub_le _ _).trans (by have h := supportedSmoothing_norm_bound Q hε (K*ε) (norm_nonneg f) f.norm_coe_le_norm z linarith [f.norm_coe_le_norm z]) · filter_upwards [ae_double_position_in_table Q] with z hz simpa only [sub_self, norm_zero, OfNat.ofNat_ne_zero, ne_eq, not_false_eq_true, zero_pow] using (((supported_smoothing_pointwise Q K hK f z hz).sub (tendsto_const_nhds (x := f z))).norm.pow 2) simp only [integral_zero] at hi have he (ε : ℝ) : lpNorm (supportedSmoothing Q ε (K*ε) f - (f : DoublePhase → ℂ)) 2 (doubleMeasure Q) = Real.sqrt (∫ z, ‖supportedSmoothing Q ε (K*ε) f z - f z‖^2 ∂doubleMeasure Q) := by rw [lpNorm_eq_integral_norm_rpow_toReal (by norm_num) (by norm_num) ((supportedSmoothing_stronglyMeasurable Q ε (K*ε) f.continuous.stronglyMeasurable).sub f.continuous.stronglyMeasurable).aestronglyMeasurable] simp only [ENNReal.toReal_ofNat, Pi.sub_apply, Real.rpow_two, Real.sqrt_eq_rpow] norm_num simp_rw [he] simpa only [Real.sqrt_zero, Function.comp_def] using Real.continuous_sqrt.continuousAt.tendsto.comp hi lemma directS_integrable_ae (Q : Triangle) {ε : ℝ} (hε : 0 < ε) {f : DoublePhase → ℂ} (hi : Integrable f (doubleMeasure Q)) : ∀ᵐ z ∂doubleMeasure Q, Integrable (fun y => kernel ε (z.1.1-y) • f ((y,z.1.2),z.2)) (volume.restrict Q.table) := by obtain ⟨C,hC⟩ := (kernel_hasCompactSupport hε).exists_bound_of_continuous (kernel_contDiff ε).continuous filter_upwards [integrable_slice_ae Q hi] with z hz exact hz.bdd_smul C (((kernel_contDiff ε).continuous.comp (continuous_const.sub continuous_id)).aestronglyMeasurable) (Eventually.of_forall fun y => hC (z.1.1-y)) lemma reflectedS_integrable_ae (Q : Triangle) (i : Fin 3) {ε : ℝ} (hε : 0 < ε) {f : DoublePhase → ℂ} (hi : Integrable f (doubleMeasure Q)) : ∀ᵐ z ∂doubleMeasure Q, Integrable (fun y => kernel ε (z.1.1-wallReflection Q i y) • f ((y,reflectedDirection Q i z.1.2),z.2+1)) (volume.restrict Q.table) := by obtain ⟨C,hC⟩ := (kernel_hasCompactSupport hε).exists_bound_of_continuous (kernel_contDiff ε).continuous filter_upwards [(measurePreserving_directionParityReflection Q i).quasiMeasurePreserving.ae (integrable_slice_ae Q hi)] with z hz exact hz.bdd_smul C (((kernel_contDiff ε).continuous.comp (continuous_const.sub (wallReflection_contDiff Q i).continuous)).aestronglyMeasurable) (Eventually.of_forall fun y => hC (z.1.1-wallReflection Q i y)) lemma smoothing_sub_ae (Q : Triangle) {ε : ℝ} (hε : 0 < ε) {f g : DoublePhase → ℂ} (hfi : Integrable f (doubleMeasure Q)) (hgi : Integrable g (doubleMeasure Q)) : reflectedSmoothing Q ε (f-(g : DoublePhase → ℂ)) =ᵐ[doubleMeasure Q] reflectedSmoothing Q ε f - reflectedSmoothing Q ε g := by filter_upwards [directS_integrable_ae Q hε hfi, directS_integrable_ae Q hε hgi, Filter.eventually_all.mpr (fun i => reflectedS_integrable_ae Q i hε hfi), Filter.eventually_all.mpr (fun i => reflectedS_integrable_ae Q i hε hgi)] with z hf hg hrf hrg have hd : directS Q ε (f-(g : DoublePhase → ℂ)) z.2 z.1.2 z.1.1 = directS Q ε f z.2 z.1.2 z.1.1 - directS Q ε g z.2 z.1.2 z.1.1 := by simp only [directS, Pi.sub_apply, smul_sub] exact integral_sub hf hg have hr (i : Fin 3) : reflectedS Q i ε (f-(g : DoublePhase → ℂ)) z.2 z.1.2 z.1.1 = reflectedS Q i ε f z.2 z.1.2 z.1.1 - reflectedS Q i ε g z.2 z.1.2 z.1.1 := by simp only [reflectedS, Pi.sub_apply, smul_sub] exact integral_sub (hrf i) (hrg i) simp only [Pi.sub_apply, reflectedSmoothing, hd, hr, Finset.sum_sub_distrib] abel lemma supported_smoothing_sub_ae (Q : Triangle) {ε : ℝ} (hε : 0 < ε) (R : ℝ) {f g : DoublePhase → ℂ} (hfi : Integrable f (doubleMeasure Q)) (hgi : Integrable g (doubleMeasure Q)) : supportedSmoothing Q ε R (f-(g : DoublePhase → ℂ)) =ᵐ[doubleMeasure Q] supportedSmoothing Q ε R f - supportedSmoothing Q ε R g := by filter_upwards [smoothing_sub_ae Q hε hfi hgi] with z hz simp only [Pi.sub_apply] at hz ⊢ simp only [supportedSmoothing, hz, smul_sub] lemma supportedSmoothing_norm_le (Q : Triangle) (ε R : ℝ) (f : DoublePhase → ℂ) (z : DoublePhase) : ‖supportedSmoothing Q ε R f z‖ ≤ ‖reflectedSmoothing Q ε f z‖ := by rw [supportedSmoothing, norm_smul, Real.norm_eq_abs, abs_of_nonneg (vertexCutoff_mem_Icc Q R _).1] simpa only [one_mul] using mul_le_mul_of_nonneg_right (vertexCutoff_mem_Icc Q R _).2 (norm_nonneg _) lemma supportedSmoothing_memLp_of_memLp (Q : Triangle) {ε : ℝ} (hε : 0 < ε) (R : ℝ) {f : DoublePhase → ℂ} (hf : StronglyMeasurable f) (hf₂ : MemLp f 2 (doubleMeasure Q)) : MemLp (supportedSmoothing Q ε R f) 2 (doubleMeasure Q) := (reflectedSmoothing_memLp Q hε hf hf₂).norm.mono' (supportedSmoothing_stronglyMeasurable Q ε R hf).aestronglyMeasurable (Eventually.of_forall (supportedSmoothing_norm_le Q ε R f)) lemma supported_smoothing_lpNorm_bound (Q : Triangle) {ε : ℝ} (hε : 0 < ε) (R : ℝ) {f : DoublePhase → ℂ} (hf : StronglyMeasurable f) (hf₂ : MemLp f 2 (doubleMeasure Q)) : lpNorm (supportedSmoothing Q ε R f) 2 (doubleMeasure Q) ≤ 4 * lpNorm f 2 (doubleMeasure Q) := by rw [← toReal_eLpNorm, ← toReal_eLpNorm] have hh : eLpNorm (supportedSmoothing Q ε R f) 2 (doubleMeasure Q) ≤ 4 * eLpNorm f 2 (doubleMeasure Q) := (eLpNorm_mono_ae (supportedSmoothing_stronglyMeasurable Q ε R hf).aestronglyMeasurable (Eventually.of_forall (supportedSmoothing_norm_le Q ε R f))).trans (reflectedSmoothing_eLpNorm Q hε hf) simpa only [ENNReal.toReal_mul, ENNReal.toReal_ofNat] using ENNReal.toReal_mono (ENNReal.mul_ne_top (by norm_num) hf₂.eLpNorm_ne_top) hh end TriangularBilliards namespace TriangularBilliards open Analysis SpatialSmoothing MeasureTheory Filter Set open scoped Topology ContDiff NNReal BoundedContinuousFunction /-- Strong L2 approximate identity for the actual supported triangular mollifier. No smoothness or invariance is required of the input. -/ lemma supported_smoothing_lpNorm_tendsto (Q : Triangle) {K : ℝ} (hK : 0 < K) {f : DoublePhase → ℂ} (hf : StronglyMeasurable f) (hf₂ : MemLp f 2 (doubleMeasure Q)) : Tendsto (fun ε : ℝ => lpNorm (supportedSmoothing Q ε (K*ε) f - (f : DoublePhase → ℂ)) 2 (doubleMeasure Q)) (𝓝[>] 0) (𝓝 0) := by apply Metric.tendsto_nhds.mpr intro δ hδ let η : ℝ := δ/12 have hη : 0 < η := by dsimp [η]; positivity obtain ⟨g,hg,hg₂⟩ := hf₂.exists_boundedContinuous_eLpNorm_sub_le (by norm_num : (2 : ℝ≥0∞) ≠ ⊤) (ENNReal.ofReal_ne_zero_iff.mpr hη) have hsmall : lpNorm (f - (g : DoublePhase → ℂ)) 2 (doubleMeasure Q) ≤ η := by rw [← toReal_eLpNorm] exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hg).trans_eq (ENNReal.toReal_ofReal hη.le) have hgsmall := (Metric.tendsto_nhds.mp (supported_smoothing_lpNorm_tendsto_bounded Q hK g)) (δ/2) (by positivity) filter_upwards [self_mem_nhdsWithin, hgsmall] with ε hε he let Sf := supportedSmoothing Q ε (K*ε) f let Sg := supportedSmoothing Q ε (K*ε) g have hfm := supportedSmoothing_memLp_of_memLp Q hε (K*ε) hf hf₂ have hgm := supportedSmoothing_memLp_of_memLp Q hε (K*ε) g.continuous.stronglyMeasurable hg₂ have hdm := supportedSmoothing_memLp_of_memLp Q hε (K*ε) (hf.sub g.continuous.stronglyMeasurable) (hf₂.sub hg₂) have hae : Sf - Sg =ᵐ[doubleMeasure Q] supportedSmoothing Q ε (K*ε) (f-(g : DoublePhase → ℂ)) := (supported_smoothing_sub_ae Q hε (K*ε) (hf₂.integrable (by norm_num)) (hg₂.integrable (by norm_num))).symm have hb : lpNorm (Sf-Sg) 2 (doubleMeasure Q) ≤ 4*η := by rw [lpNorm_congr_ae_of_strong (hfm.sub hgm).aestronglyMeasurable hdm.aestronglyMeasurable hae] exact (supported_smoothing_lpNorm_bound Q hε (K*ε) (hf.sub g.continuous.stronglyMeasurable) (hf₂.sub hg₂)).trans (by gcongr) have hn : lpNorm (Sf-f) 2 (doubleMeasure Q) ≤ lpNorm (Sf-Sg) 2 (doubleMeasure Q) + lpNorm (Sg-(g : DoublePhase → ℂ)) 2 (doubleMeasure Q) + lpNorm (f-(g : DoublePhase → ℂ)) 2 (doubleMeasure Q) := by have heq : Sf-f = ((Sf-Sg)+(Sg-(g : DoublePhase → ℂ)))-(f-(g : DoublePhase → ℂ)) := by ext z; simp only [Pi.sub_apply, Pi.add_apply]; abel rw [heq] apply (lpNorm_sub_le ((hfm.sub hgm).add (hgm.sub hg₂)) (by norm_num)).trans exact add_le_add (lpNorm_add_le (hfm.sub hgm) (by norm_num)) le_rfl rw [Real.dist_eq, sub_zero, abs_of_nonneg lpNorm_nonneg] at he ⊢ change lpNorm (Sf - f) 2 (doubleMeasure Q) < δ change lpNorm (Sg - (g : DoublePhase → ℂ)) 2 (doubleMeasure Q) < δ/2 at he dsimp [η] at hsmall hb linarith end TriangularBilliards end end end end end end end end end end OAI