| import Mathlib |
| import OAI.MeasureTheory.FalconerCompletion.Fourier.FourierPolynomialSmoothing |
|
|
| namespace OAI |
|
|
| /-! Angular cutoffs and derivative estimates. -/ |
|
|
| section |
| section |
| namespace GraphLiteralSupport.CurrentFalconer.Exceptional |
| open Set Metric MeasureTheory |
| open scoped ENNReal RealInnerProductSpace |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
|
|
| lemma cone_unit_cap {z w : V} (hz : ‖z‖=1) (hw : ‖w‖=1) : |
| dist w z ≤ 2*‖graphPerpND w z‖ ∨ dist w (-z) ≤ 2*‖graphPerpND w z‖ := by |
| have hi : |inner ℝ z w| ≤ 1 := by simpa [hz,hw] using abs_real_inner_le_norm z w |
| have hp : ‖graphPerpND w z‖^2 = 1-(inner ℝ z w)^2 := by |
| rw [graphPerpND,norm_sub_sq_real,norm_smul,inner_smul_right,hz,hw,Real.norm_eq_abs] |
| simp only [mul_one,one_pow,sq_abs] |
| ring |
| have hd : dist w z ^ 2 = 2-2*inner ℝ z w := by |
| rw [dist_eq_norm,norm_sub_sq_real,hz,hw,real_inner_comm] |
| ring |
| have hd' : dist w (-z) ^ 2 = 2+2*inner ℝ z w := by |
| rw [dist_eq_norm,norm_sub_sq_real,norm_neg,hz,hw,inner_neg_right,real_inner_comm] |
| ring |
| rcases le_total 0 (inner ℝ z w) with hs | hs |
| · left |
| have hh := (abs_le.mp hi).2 |
| nlinarith [sq_nonneg (1-inner ℝ z w),dist_nonneg (x:=w) (y:=z),norm_nonneg (graphPerpND w z), |
| mul_nonneg hs (sub_nonneg.mpr hh)] |
| · right |
| have hh := (abs_le.mp hi).1 |
| nlinarith [sq_nonneg (1+inner ℝ z w),dist_nonneg (x:=w) (y:=(-z)),norm_nonneg (graphPerpND w z), |
| mul_nonpos_of_nonpos_of_nonneg hs (by linarith : 0 ≤ 1+inner ℝ z w)] |
|
|
| lemma cone_cap_cover {v : V} (hv : v ≠ 0) {W : ℝ} (hW : 0 < W) : |
| {w | ‖w‖=1 ∧ ‖graphPerpND w v‖ ≤ W} ⊆ |
| ball (‖v‖⁻¹ • v) (3*W/‖v‖) ∪ ball (-(‖v‖⁻¹ • v)) (3*W/‖v‖) := by |
| intro w hw |
| have hnv : 0 < ‖v‖ := norm_pos_iff.mpr hv |
| have hz : ‖‖v‖⁻¹ • v‖ = 1 := by |
| rw [norm_smul,Real.norm_eq_abs,abs_inv,abs_norm,inv_mul_cancel₀ hnv.ne'] |
| have he : graphPerpND w (‖v‖⁻¹ • v) = ‖v‖⁻¹ • graphPerpND w v := by |
| simp only [graphPerpND,real_inner_smul_left,smul_sub,smul_smul] |
| have hb : 2*‖graphPerpND w (‖v‖⁻¹ • v)‖ < 3*W/‖v‖ := by |
| rw [he,norm_smul,Real.norm_eq_abs,abs_inv,abs_norm] |
| have hh := mul_le_mul_of_nonneg_left hw.2 (show 0 ≤ ‖v‖⁻¹ by positivity) |
| have hp : 0 < W/‖v‖ := by positivity |
| simp only [div_eq_mul_inv] at * |
| nlinarith |
| rcases cone_unit_cap hz hw.1 with hc | hc |
| · exact Or.inl (mem_ball.mpr (hc.trans_lt hb)) |
| · exact Or.inr (mem_ball.mpr (hc.trans_lt hb)) |
| end GraphLiteralSupport.CurrentFalconer.Exceptional |
| end |
| end |
|
|
| section |
| section |
| namespace GraphLiteralSupport.CurrentFalconer.Exceptional |
| open Set Metric MeasureTheory |
| open scoped ENNReal RealInnerProductSpace |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [MeasurableSpace V] |
|
|
| omit [InnerProductSpace ℝ V] in |
| lemma cap_mass_large_radius (ξ : Measure V) {L S R r : ℝ} (hL : 1 ≤ L) (hS : 0 ≤ S) |
| (hmass : ξ univ ≤ 1) (hc : ∀ w : V, ‖w‖=1 → ∀ r : ℝ, R⁻¹ ≤ r → r ≤ 1 → |
| ξ (ball w r) ≤ ENNReal.ofReal (L*r^S)) (w : V) (hw : ‖w‖=1) (hr : R⁻¹ ≤ r) : |
| ξ (ball w r) ≤ ENNReal.ofReal (L*r^S) := by |
| by_cases hr1 : r ≤ 1 |
| · exact hc w hw r hr hr1 |
| · have hh : 1 ≤ L*r^S := one_le_mul_of_one_le_of_one_le hL (Real.one_le_rpow (by linarith) hS) |
| exact (measure_mono (subset_univ _)).trans (hmass.trans (by exact_mod_cast ENNReal.ofReal_le_ofReal hh)) |
|
|
| lemma cone_mass_bound (ξ : Measure V) {L S R r W : ℝ} (hL : 1 ≤ L) (hS : 0 ≤ S) |
| (hmass : ξ univ ≤ 1) (hsphere : ∀ᵐ w ∂ξ, ‖w‖=1) |
| (hc : ∀ w : V, ‖w‖=1 → ∀ r : ℝ, R⁻¹ ≤ r → r ≤ 1 → |
| ξ (ball w r) ≤ ENNReal.ofReal (L*r^S)) (v : V) (hv : v ≠ 0) (hW : 0 < W) |
| (hr : R⁻¹ ≤ r) (hr0 : 0 ≤ r) (hvr : 3*W/‖v‖ ≤ r) : |
| ξ {w | ‖graphPerpND w v‖ ≤ W} ≤ ENNReal.ofReal (2*L*r^S) := by |
| have hz : ‖‖v‖⁻¹ • v‖=1 := by |
| rw [norm_smul,Real.norm_eq_abs,abs_inv,abs_norm,inv_mul_cancel₀ (norm_ne_zero_iff.mpr hv)] |
| have hzn : ‖-(‖v‖⁻¹ • v)‖=1 := by simpa using hz |
| calc |
| _ ≤ ξ (ball (‖v‖⁻¹ • v) r ∪ ball (-(‖v‖⁻¹ • v)) r) := by |
| apply measure_mono_ae |
| filter_upwards [hsphere] with w hw |
| intro hp |
| exact union_subset_union (ball_subset_ball hvr) (ball_subset_ball hvr) (cone_cap_cover hv hW ⟨hw,hp⟩) |
| _ ≤ ξ (ball (‖v‖⁻¹ • v) r)+ξ (ball (-(‖v‖⁻¹ • v)) r) := measure_union_le _ _ |
| _ ≤ ENNReal.ofReal (L*r^S)+ENNReal.ofReal (L*r^S) := add_le_add |
| (cap_mass_large_radius ξ hL hS hmass hc _ hz hr) |
| (cap_mass_large_radius ξ hL hS hmass hc _ hzn hr) |
| _ = _ := by rw [←ENNReal.ofReal_add (by positivity) (by positivity)]; congr 1; ring |
| end GraphLiteralSupport.CurrentFalconer.Exceptional |
| end |
| end |
|
|
| end OAI |
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|
