| import Mathlib |
| import OAI.Geometry.PrescribedPotential.CircleRadialCalculus |
| import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives |
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| namespace OAI |
|
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| /-! Ball Submean. -/ |
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| section |
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| |
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| noncomputable section |
| open Set Filter Topology Metric MeasureTheory |
| open scoped ContDiff InnerProductSpace |
| namespace PotentialABP |
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| section Affine |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
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| lemma hessian_affine_comp {f : F → ℝ} (hf : ContDiff ℝ ∞ f) |
| (L : E →L[ℝ] F) (c : F) (z u v : E) : |
| fderiv ℝ (fderiv ℝ (fun y => f (c + L y))) z u v = |
| fderiv ℝ (fderiv ℝ f) (c + L z) (L u) (L v) := by |
| have hd : fderiv ℝ (fun y => c + L y) = fun _ => L := by |
| funext y |
| exact (L.hasFDerivAt.const_add c).fderiv |
| have hc : ContDiff ℝ ∞ (fun y => c + L y) := contDiff_const.add L.contDiff |
| change fderiv ℝ (fderiv ℝ (f ∘ (fun y => c + L y))) z u v = _ |
| rw [PotentialKaehler.second_derivative_comp hc.contDiffAt hf.contDiffAt, hd] |
| simp |
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| end Affine |
|
|
| section ComplexHilbert |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] |
| [FiniteDimensional ℂ E] [MeasurableSpace E] [BorelSpace E] |
| local instance ballSubmeanRealInnerProductSpace : InnerProductSpace ℝ E := |
| InnerProductSpace.rclikeToReal ℂ E |
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| private def unitScalarIso (a : ℂ) (ha : ‖a‖ = 1) : E ≃ₗᵢ[ℝ] E where |
| toLinearEquiv := (LinearEquiv.smulOfNeZero ℂ E a (by |
| intro h; simp [h] at ha)).restrictScalars ℝ |
| norm_map' x := by simp [LinearEquiv.smulOfNeZero_apply, norm_smul, ha] |
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| lemma ball_integral_unit_smul (f : E → ℝ) (r : ℝ) (a : ℂ) (ha : ‖a‖ = 1) : |
| ∫ x in ball (0:E) r, f (a • x) = ∫ x in ball (0:E) r, f x := by |
| let e : E ≃ₗᵢ[ℝ] E := unitScalarIso a ha |
| have hb : e ⁻¹' ball (0:E) r = ball (0:E) r := by |
| ext x |
| simp only [mem_preimage, mem_ball, dist_zero_right, e.norm_map] |
| have hm := e.measurePreserving.restrict_preimage (s := ball (0:E) r) isOpen_ball.measurableSet |
| rw [hb] at hm |
| exact hm.integral_comp e.toHomeomorph.measurableEmbedding f |
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| lemma ball_integral_translate (f : E → ℝ) (c : E) (r : ℝ) : |
| ∫ x in ball (0:E) r, f (c+x) = ∫ x in ball c r, f x := by |
| have hb : (fun x : E => c+x) ⁻¹' ball c r = ball (0:E) r := by |
| ext x |
| simp [mem_ball, dist_eq_norm] |
| have hm := (measurePreserving_add_left (volume : Measure E) c).restrict_preimage |
| (s := ball c r) isOpen_ball.measurableSet |
| rw [hb] at hm |
| exact hm.integral_comp (Homeomorph.addLeft c).measurableEmbedding f |
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| omit [FiniteDimensional ℂ E] [MeasurableSpace E] [BorelSpace E] in |
| lemma complexLine_hessian {f : E → ℝ} (hf : ContDiff ℝ ∞ f) |
| (c v : E) (z a b : ℂ) : |
| fderiv ℝ (fderiv ℝ (fun w : ℂ => f (c + w • v))) z a b = |
| fderiv ℝ (fderiv ℝ f) (c + z • v) (a • v) (b • v) := by |
| let L : ℂ →L[ℝ] E := ((ContinuousLinearMap.id ℂ ℂ).smulRight v).restrictScalars ℝ |
| exact hessian_affine_comp hf L c z a b |
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| |
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| theorem smooth_ball_submean {f : E → ℝ} (hf : ContDiff ℝ ∞ f) |
| (c : E) {R : ℝ} (hR : 0 ≤ R) |
| (hlevi : ∀ x ∈ closedBall c R, ∀ v : E, |
| 0 ≤ fderiv ℝ (fderiv ℝ f) x v v + |
| fderiv ℝ (fderiv ℝ f) x (Complex.I • v) (Complex.I • v)) : |
| (volume.real (ball (0:E) R)) * f c ≤ ∫ x in ball c R, f x := by |
| let F : ℝ → E → ℝ := fun θ v => f (c + circleMap 0 1 θ • v) |
| have hF : Continuous (Function.uncurry F) := hf.continuous.comp |
| (continuous_const.add (((continuous_circleMap 0 1).comp continuous_fst).smul continuous_snd)) |
| have hi : Integrable (Function.uncurry F) |
| ((volume.restrict (uIoc 0 (2*Real.pi))).prod |
| (volume.restrict (ball (0:E) R))) := by |
| rw [Measure.prod_restrict] |
| apply (hF.continuousOn.integrableOn_compact |
| (isCompact_uIcc.prod (isCompact_closedBall (0:E) R))).mono_set |
| exact prod_mono uIoc_subset_uIcc ball_subset_closedBall |
| have hv (v : E) (hv : v ∈ ball (0:E) R) : |
| 2*Real.pi*f c ≤ ∫ θ in 0..2*Real.pi, F θ v := by |
| have hc : ContDiff ℝ ∞ (fun z : ℂ => f (c + z • v)) := |
| hf.comp (contDiff_const.add (contDiff_id.smul contDiff_const)) |
| have h := smooth_circle_submean hc 0 (by norm_num : (0:ℝ) ≤ 1) ?_ |
| · simp only [zero_smul, add_zero] at h |
| unfold Real.circleAverage at h |
| simp only [smul_eq_mul] at h |
| exact (le_inv_mul_iff₀ (by positivity : 0 < 2*Real.pi)).mp h |
| · intro z hz |
| rw [complexLine_hessian hf, complexLine_hessian hf, one_smul] |
| apply hlevi |
| have hz' : ‖z‖ ≤ 1 := by simpa [mem_closedBall] using hz |
| have hv' : ‖v‖ ≤ R := (by simpa [mem_ball] using hv : ‖v‖ < R).le |
| simp only [mem_closedBall, dist_eq_norm, add_sub_cancel_left, norm_smul] |
| exact (mul_le_mul_of_nonneg_left hv' (norm_nonneg z)).trans |
| (mul_le_of_le_one_left hR hz') |
| have hmean : (2*Real.pi) * (volume.real (ball (0:E) R) * f c) ≤ |
| ∫ v in ball (0:E) R, ∫ θ in 0..2*Real.pi, F θ v := by |
| have hi' : IntegrableOn (fun v => ∫ θ in 0..2*Real.pi, F θ v) (ball (0:E) R) := by |
| simpa only [intervalIntegral.integral_of_le (by positivity : (0:ℝ) ≤ 2*Real.pi), |
| uIoc_of_le (by positivity : (0:ℝ) ≤ 2*Real.pi), Function.uncurry_def, |
| IntegrableOn] using hi.integral_prod_right |
| have hiC : IntegrableOn (fun _ : E => 2*Real.pi*f c) (ball (0:E) R) := |
| (continuous_const.continuousOn.integrableOn_compact (isCompact_closedBall (0:E) R)).mono_set |
| ball_subset_closedBall |
| have hm := setIntegral_mono_on hiC hi' isOpen_ball.measurableSet hv |
| simpa only [setIntegral_const, smul_eq_mul, mul_left_comm] using hm |
| rw [← intervalIntegral_integral_swap hi] at hmean |
| have he (θ : ℝ) : (∫ v in ball (0:E) R, F θ v) = ∫ x in ball c R, f x := by |
| exact (ball_integral_unit_smul (fun x => f (c+x)) R (circleMap 0 1 θ) |
| (by simp)).trans (ball_integral_translate f c R) |
| simp_rw [he] at hmean |
| simp only [intervalIntegral.integral_const, sub_zero, smul_eq_mul] at hmean |
| nlinarith [Real.pi_pos] |
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|
| end ComplexHilbert |
| end PotentialABP |
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| end |
| end |
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| end OAI |
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