| import OAI.Geometry.ProjectionBodies.ScalarCalculus |
|
|
| namespace OAI |
|
|
| universe uE |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology Function |
| open scoped ENNReal RealInnerProductSpace |
| namespace PettyProjection.Spherical |
|
|
| lemma mean_add {n : ℕ} [NeZero n] {f g : Sphere n → ℝ} (hf : Continuous f) (hg : Continuous g) : |
| mean (fun u => f u + g u) = mean f + mean g := |
| integral_add (continuous_integrable hf) (continuous_integrable hg) |
|
|
| lemma mean_sub {n : ℕ} [NeZero n] {f g : Sphere n → ℝ} (hf : Continuous f) (hg : Continuous g) : |
| mean (fun u => f u - g u) = mean f - mean g := |
| integral_sub (continuous_integrable hf) (continuous_integrable hg) |
|
|
| lemma mean_const_mul {n : ℕ} (c : ℝ) (f : Sphere n → ℝ) : |
| mean (fun u => c * f u) = c * mean f := integral_const_mul c f |
|
|
| lemma mean_mul_const {n : ℕ} (c : ℝ) (f : Sphere n → ℝ) : |
| mean (fun u => f u * c) = mean f * c := integral_mul_const c f |
|
|
| lemma mean_div_const {n : ℕ} (c : ℝ) (f : Sphere n → ℝ) : |
| mean (fun u => f u / c) = mean f / c := integral_div c f |
|
|
| lemma mean_mono {n : ℕ} [NeZero n] {f g : Sphere n → ℝ} (hf : Continuous f) (hg : Continuous g) |
| (h : ∀ u, f u ≤ g u) : mean f ≤ mean g := |
| integral_mono (continuous_integrable hf) (continuous_integrable hg) h |
|
|
| lemma mean_nonneg {n : ℕ} {f : Sphere n → ℝ} (h : ∀ u, 0 ≤ f u) : 0 ≤ mean f := |
| integral_nonneg h |
|
|
| lemma mean_centered_square {n : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) : |
| mean (fun u => (f u - mean f) ^ 2) = variance f := by |
| have hi : (fun u => (f u - mean f) ^ 2) = |
| (fun u => (f u) ^ 2 - 2 * mean f * f u + (mean f) ^ 2) := by |
| funext u; ring |
| rw [hi, mean_add (f := fun u => (f u)^2 - 2 * mean f * f u) |
| (g := fun _ => (mean f)^2) |
| ((f.continuous.pow 2).sub (continuous_const.mul f.continuous)) continuous_const, |
| mean_sub (f := fun u => (f u)^2) (g := fun u => 2 * mean f * f u) |
| (f.continuous.pow 2) (continuous_const.mul f.continuous), |
| mean_const_mul, mean_const, variance] |
| ring |
|
|
| lemma norm_Q_le_variance {n : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) : |
| ‖Q n (toH n f)‖ ^ 2 ≤ variance f := by |
| simpa only [mean_centered_square] using norm_Q_le_centered n f (mean f) |
|
|
| lemma weighted_square_bound {a b w : ℝ} (hw : 0 < w) (hw1 : w < 1) : |
| (a+b)^2 ≤ a^2/w + b^2/(1-w) := by |
| have h1 : 0 < 1-w := sub_pos.mpr hw1 |
| apply (mul_le_mul_iff_of_pos_right (mul_pos hw h1)).mp |
| have he : (a^2/w + b^2/(1-w)) * (w*(1-w)) = a^2*(1-w)+b^2*w := by |
| field_simp |
| rw [he] |
| nlinarith only [sq_nonneg ((1-w)*a-w*b)] |
|
|
| lemma weighted_norm_bound {E : Type uE} [SeminormedAddCommGroup E] (a b : E) |
| {w : ℝ} (hw : 0 < w) (hw1 : w < 1) : |
| ‖a+b‖^2 ≤ ‖a‖^2/w + ‖b‖^2/(1-w) := |
| (sq_le_sq₀ (norm_nonneg _) (add_nonneg (norm_nonneg _) (norm_nonneg _))).mpr |
| (norm_add_le a b) |>.trans (weighted_square_bound hw hw1) |
|
|
| lemma scalar_correction {n : ℕ} (hn : 4 ≤ n) {m t : ℝ} |
| (hm : 0 ≤ m) (ht : 0 ≤ t) (htb : t ≤ Scalar.beta n * m) : |
| Scalar.coefficient n * ((2*n : ℝ) * t^2 / (Scalar.weight n * (2*n+8)) - |
| (m-t)^2/(1-Scalar.weight n)) ≤ |
| (2*(Scalar.exponent n : ℝ)-1)/(Scalar.exponent n : ℝ)^2 * m^2 := by |
| have hb := (Scalar.beta_pos n).le |
| have hb1 := (Scalar.beta_lt_one n).le |
| have hw := Scalar.weight_pos n |
| have hw1 := Scalar.weight_lt_one n |
| have hden : 0 < Scalar.weight n * (2*n+8 : ℝ) := mul_pos hw (by positivity) |
| have hs1 : t^2 ≤ (Scalar.beta n)^2 * m^2 := by |
| simpa only [mul_pow] using (sq_le_sq₀ ht (mul_nonneg hb hm)).mpr htb |
| have ht1 : t ≤ m := htb.trans (by nlinarith) |
| have hs2 : (1-Scalar.beta n)^2*m^2 ≤ (m-t)^2 := by |
| have hle : (1-Scalar.beta n)*m ≤ m-t := by nlinarith |
| simpa only [mul_pow] using (sq_le_sq₀ (mul_nonneg (sub_nonneg.mpr hb1) hm) |
| (sub_nonneg.mpr ht1)).mpr hle |
| have h1 := div_le_div_of_nonneg_right |
| (mul_le_mul_of_nonneg_left hs1 (by positivity : 0 ≤ (2*n : ℝ))) hden.le |
| have h2 := div_le_div_of_nonneg_right hs2 (sub_nonneg.mpr hw1.le) |
| have h3 := mul_le_mul_of_nonneg_left (sub_le_sub h1 h2) (Scalar.coefficient_pos n).le |
| have h4 := mul_le_mul_of_nonneg_right (Scalar.mean_bound n hn) (sq_nonneg m) |
| apply h3.trans |
| convert h4 using 1; ring |
|
|
| lemma scalar_curvature {n : ℕ} (hn : 4 ≤ n) [NeZero n] (g : Seminorm ℝ (Space n)) : |
| (2 * n : ℝ) * variance (fun u : Sphere n => Scalar.G n (g u)) + (2 * n + 8 : ℝ) * |
| ‖Q n (toH n (restrictContinuous (Scalar.G n ∘ g) |
| ((Scalar.contDiff_G hn).continuous.comp (seminorm_continuous n g))))‖ ^ 2 ≤ |
| ((n : ℝ) - 1) * Scalar.beta n ^ 2 * |
| mean (fun u : Sphere n => Scalar.I n (g u) * g u) := |
| seminorm_curvature (by omega) g (Scalar.contDiff_G hn) (Scalar.contDiff_I hn) |
| (Scalar.I_nonneg hn) (Scalar.derivative_relation hn) |
|
|
| end PettyProjection.Spherical |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology Function |
| open scoped ENNReal RealInnerProductSpace |
| namespace PettyProjection.Spherical |
| open Scalar |
|
|
| lemma continuous_e_comp {n : ℕ} (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) : |
| Continuous (fun u => e n (f u)) := |
| (((f.continuous.inv₀ (fun u => (hf u).ne')).pow n).sub continuous_const).div_const _ |
| |
| lemma continuous_M_comp {n : ℕ} (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) : |
| Continuous (fun u => M n (f u)) := |
| ((f.continuous.pow _).sub continuous_const).add |
| (continuous_const.mul (continuous_e_comp f hf)) |
| |
| lemma mean_e_zero {n : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) |
| (hnorm : mean (fun u => (f u)⁻¹ ^ n) = 1) : mean (fun u => e n (f u)) = 0 := by |
| unfold e |
| rw [mean_div_const, mean_sub (f := fun u => (f u)⁻¹^n) (g := fun _ => 1) ((f.continuous.inv₀ (fun u => (hf u).ne')).pow n) |
| continuous_const, mean_const, hnorm, sub_self, zero_div] |
|
|
| lemma mean_M_of_normalized {n : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) |
| (hnorm : mean (fun u => (f u)⁻¹ ^ n) = 1) : |
| mean (fun u => M n (f u)) = mean (fun u => (f u) ^ exponent n) - 1 := by |
| unfold M |
| rw [mean_add (f := fun u => (f u)^exponent n - 1) |
| (g := fun u => (exponent n : ℝ) * e n (f u)) |
| ((f.continuous.pow _).sub continuous_const) (continuous_const.mul (continuous_e_comp f hf)), |
| mean_sub (f := fun u => (f u)^exponent n) (g := fun _ => 1) (f.continuous.pow _) continuous_const, mean_const, mean_const_mul, |
| mean_e_zero f hf hnorm, mul_zero, add_zero] |
|
|
| lemma mean_W_of_normalized {n : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) |
| (hnorm : mean (fun u => (f u)⁻¹ ^ n) = 1) : |
| mean (fun u => W n (f u)) = mean (fun u => (f u)^exponent n) - |
| mean (fun u => (f u)^(exponent n-1)) := by |
| unfold W |
| rw [mean_add (f := fun u => (f u)^exponent n - (f u)^(exponent n-1)) |
| (g := fun u => e n (f u)) ((f.continuous.pow _).sub (f.continuous.pow _)) |
| (continuous_e_comp f hf), |
| mean_sub (f := fun u => (f u)^exponent n) (g := fun u => (f u)^(exponent n-1)) (f.continuous.pow _) (f.continuous.pow _), mean_e_zero f hf hnorm, add_zero] |
|
|
| lemma mean_first_power_ge_one {n : ℕ} (hn : 0 < n) [NeZero n] |
| (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) |
| (hnorm : mean (fun u => (f u)⁻¹ ^ n) = 1) : 1 ≤ mean f := by |
| have hi := mean_mono (f := fun u => 1-f u) (g := fun u => e n (f u)) (continuous_const.sub f.continuous) (continuous_e_comp f hf) |
| (fun u => e_lower hn (hf u)) |
| rw [mean_sub (f := fun _ => 1) (g := f) continuous_const f.continuous, mean_const, mean_e_zero f hf hnorm] at hi |
| linarith |
|
|
| lemma norm_mean_ranges {n : ℕ} (hn : 4 ≤ n) [NeZero n] |
| (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) |
| (hnorm : mean (fun u => (f u)⁻¹ ^ n) = 1) : |
| let m := mean (fun u => (f u) ^ exponent n) - 1 |
| let ell := mean (fun u => (f u) ^ (exponent n - 1)) - 1 |
| let t := mean (fun u => G n (f u)) |
| 0 ≤ m ∧ 0 ≤ ell ∧ ell ≤ ((exponent n : ℝ) - 1) / exponent n * m ∧ |
| 0 ≤ t ∧ t ≤ beta n * m := by |
| dsimp only |
| have hm : 0 ≤ mean (fun u => (f u)^exponent n) - 1 := by |
| rw [← mean_M_of_normalized f hf hnorm] |
| exact mean_nonneg (fun u => M_nonneg (by omega) (hf u)) |
| have ht : 0 ≤ mean (fun u => G n (f u)) := |
| mean_nonneg (fun u => (G_bounds (by omega) (hf u)).1) |
| have htb := mean_mono (f := fun u => G n (f u)) (g := fun u => beta n*M n (f u)) ((contDiff_G hn).continuous.comp f.continuous) |
| (continuous_const.mul (continuous_M_comp f hf)) (fun u => (G_bounds (by omega) (hf u)).2) |
| rw [mean_const_mul, mean_M_of_normalized f hf hnorm] at htb |
| refine ⟨hm, ?_, ?_, ht, htb⟩ |
| · by_cases hn4 : n = 4 |
| · simp [exponent, hn4, mean_const] |
| · simpa only [exponent, ite_eq_right hn4, Nat.reduceSub, pow_one, sub_nonneg] using |
| mean_first_power_ge_one (by omega) f hf hnorm |
| · by_cases hn4 : n = 4 |
| · simp [exponent, hn4, mean_const] |
| · simp only [exponent, ite_eq_right hn4, Nat.reduceSub, Nat.cast_ofNat, pow_one] |
| have h := mean_mono (f := fun u => 2*f u) (g := fun u => (f u)^2+1) |
| (continuous_const.mul f.continuous) ((f.continuous.pow 2).add continuous_const) |
| (fun u => by nlinarith only [sq_nonneg (f u - 1)]) |
| rw [mean_const_mul, mean_add (f := fun u =>(f u)^2) (g := fun _ => 1) (f.continuous.pow 2) continuous_const, mean_const] at h |
| linarith |
|
|
| end PettyProjection.Spherical |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology Function |
| open scoped ENNReal RealInnerProductSpace |
| namespace PettyProjection.Spherical |
| open Scalar |
|
|
| lemma continuous_D {n : ℕ} (hn : 4 ≤ n) : Continuous (D n) := |
| ((continuous_id.pow _).sub continuous_const).sub (contDiff_G hn).continuous |
|
|
| lemma continuous_L {n : ℕ} (hn : 4 ≤ n) : Continuous (L n) := |
| continuous_const.mul (((((continuous_const.mul continuous_id).mul |
| (contDiff_I hn).continuous).sub (continuous_const.mul ((contDiff_G hn).continuous.pow 2))).div_const _).add |
| (((continuous_D hn).pow 2).div_const _)) |
|
|
| lemma continuous_W_comp {n : ℕ} (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) : |
| Continuous (fun u => W n (f u)) := |
| ((f.continuous.pow _).sub (f.continuous.pow _)).add (continuous_e_comp f hf) |
|
|
| lemma mean_D {n : ℕ} (hn : 4 ≤ n) [NeZero n] (f : C(Sphere n, ℝ)) : |
| mean (fun u => D n (f u)) = mean (fun u => (f u)^exponent n) - 1 - |
| mean (fun u => G n (f u)) := by |
| unfold D |
| rw [mean_sub (f := fun u => (f u)^exponent n - 1) (g := fun u => G n (f u)) |
| ((f.continuous.pow _).sub continuous_const) ((contDiff_G hn).continuous.comp f.continuous), |
| mean_sub (f := fun u => (f u)^exponent n) (g := fun _ => 1) |
| (f.continuous.pow _) continuous_const, mean_const] |
|
|
| lemma mean_L {n : ℕ} (hn : 4 ≤ n) [NeZero n] (f : C(Sphere n, ℝ)) : |
| mean (fun u => L n (f u)) = coefficient n * |
| ((((n : ℝ)-1)*beta n^2 * mean (fun u => I n (f u)*f u) - |
| 2*n*mean (fun u => (G n (f u))^2))/(weight n*(2*n+8)) + |
| mean (fun u => (D n (f u))^2)/(1-weight n)) := by |
| have hG := (contDiff_G hn).continuous.comp f.continuous |
| have hI := (contDiff_I hn).continuous.comp f.continuous |
| have hD := (continuous_D hn).comp f.continuous |
| unfold L |
| rw [mean_const_mul, |
| mean_add (f := fun u => (((n : ℝ)-1)*beta n^2*f u*I n (f u)-2*n*(G n (f u))^2)/(weight n*(2*n+8))) |
| (g := fun u => (D n (f u))^2/(1-weight n)) |
| (((continuous_const.mul f.continuous).mul hI |>.sub (continuous_const.mul (hG.pow 2))).div_const _) |
| ((hD.pow 2).div_const _), |
| mean_div_const, mean_div_const, |
| mean_sub (f := fun u => ((n : ℝ)-1)*beta n^2*f u*I n (f u)) |
| (g := fun u => 2*n*(G n (f u))^2) |
| ((continuous_const.mul f.continuous).mul hI) (continuous_const.mul (hG.pow 2)), |
| mean_const_mul] |
| congr 2 |
| congr 1 |
| have he : (fun u : Sphere n => ((n : ℝ)-1)*beta n^2*f u*I n (f u)) = |
| (fun u => ((n : ℝ)-1)*beta n^2*(I n (f u)*f u)) := by funext u; ring |
| rw [he, mean_const_mul] |
|
|
| lemma Q_power_split {n : ℕ} (hn : 4 ≤ n) [NeZero n] (f : C(Sphere n, ℝ)) : |
| Q n (toH n ⟨fun u => (f u)^exponent n, f.continuous.pow _⟩) = |
| Q n (toH n ⟨fun u => G n (f u), (contDiff_G hn).continuous.comp f.continuous⟩) + |
| Q n (toH n ⟨fun u => D n (f u), (continuous_D hn).comp f.continuous⟩) := by |
| have he : (⟨fun u => (f u)^exponent n, f.continuous.pow _⟩ : C(Sphere n, ℝ)) = |
| ContinuousMap.const (Sphere n) 1 + |
| (⟨fun u => G n (f u), (contDiff_G hn).continuous.comp f.continuous⟩ : C(Sphere n, ℝ)) + |
| ⟨fun u => D n (f u), (continuous_D hn).comp f.continuous⟩ := by |
| ext u; simp only [ContinuousMap.add_apply, ContinuousMap.const_apply, ContinuousMap.coe_mk, D]; ring |
| rw [he, map_add, map_add, map_add, map_add, Q_constant, zero_add] |
|
|
| /-- Exact expectation reduction in the proof of the norm theorem. -/ |
| lemma norm_reduction {n : ℕ} (hn : 4 ≤ n) [NeZero n] (g : Seminorm ℝ (Space n)) : |
| coefficient n * ‖Q n (toH n (restrictContinuous (fun x => g x ^ exponent n) |
| ((seminorm_continuous n g).pow _)))‖^2 ≤ |
| mean (fun u : Sphere n => L n (g u)) + coefficient n * |
| (2*n*(mean (fun u : Sphere n => G n (g u)))^2/(weight n*(2*n+8)) - |
| (mean (fun u : Sphere n => D n (g u)))^2/(1-weight n)) := by |
| let f := restrictContinuous g (seminorm_continuous n g) |
| let F : C(Sphere n, ℝ) := ⟨fun u => (f u)^exponent n, f.continuous.pow _⟩ |
| let GG : C(Sphere n, ℝ) := ⟨fun u => G n (f u), (contDiff_G hn).continuous.comp f.continuous⟩ |
| let DD : C(Sphere n, ℝ) := ⟨fun u => D n (f u), (continuous_D hn).comp f.continuous⟩ |
| have hs : Q n (toH n F) = Q n (toH n GG) + Q n (toH n DD) := Q_power_split hn f |
| have hq := weighted_norm_bound (Q n (toH n GG)) (Q n (toH n DD)) |
| (weight_pos n) (weight_lt_one n) |
| rw [← hs] at hq |
| have hd := norm_Q_le_variance DD |
| have hc := scalar_curvature hn g |
| change (2*n : ℝ)*variance GG + (2*n+8 : ℝ)*‖Q n (toH n GG)‖^2 ≤ |
| ((n : ℝ)-1)*beta n^2 * mean (fun u => I n (f u)*f u) at hc |
| have hp : 0 < (2*n+8 : ℝ) := by positivity |
| have hcg : ‖Q n (toH n GG)‖^2 ≤ |
| (((n : ℝ)-1)*beta n^2 * mean (fun u => I n (f u)*f u) - |
| (2*n : ℝ)*variance GG)/(2*n+8) := (le_div_iff₀ hp).mpr (by linarith) |
| have hbound := hq.trans (add_le_add (div_le_div_of_nonneg_right hcg (weight_pos n).le) |
| (div_le_div_of_nonneg_right hd (sub_nonneg.mpr (weight_lt_one n).le))) |
| have hbound' := mul_le_mul_of_nonneg_left hbound (coefficient_pos n).le |
| change coefficient n * ‖Q n (toH n F)‖^2 ≤ _ |
| refine hbound'.trans_eq ?_ |
| rw [show mean (fun u : Sphere n => L n (g u)) = mean (fun u => L n (f u)) from rfl, |
| mean_L hn f] |
| change coefficient n * |
| (((((n : ℝ)-1)*beta n^2 * mean (fun u => I n (f u)*f u) - |
| (2*n : ℝ)*(mean (fun u => (G n (f u))^2) - (mean (fun u => G n (f u)))^2)) / |
| (2*n+8))/weight n + |
| (mean (fun u => (D n (f u))^2) - (mean (fun u => D n (f u)))^2)/(1-weight n)) = _ |
| dsimp only [f, restrictContinuous, ContinuousMap.coe_mk] |
| simp only [div_eq_mul_inv, mul_inv_rev] |
| ring |
|
|
| lemma mean_square_remainder {k : ℝ} (hk : 0 < k) {m ell : ℝ} |
| (hm : 0 ≤ m) (hl : 0 ≤ ell) (hkm : 0 ≤ k-1) (h : ell ≤ (k-1)/k*m) : |
| (2*k-1)/k^2*m^2 ≤ m^2-ell^2 := by |
| have hs : ell^2 ≤ ((k-1)/k*m)^2 := |
| (sq_le_sq₀ hl (mul_nonneg (div_nonneg hkm hk.le) hm)).mpr h |
| have he : (2*k-1)/k^2*m^2 = m^2-((k-1)/k*m)^2 := by field_simp; ring |
| rw [he]; linarith |
|
|
| end PettyProjection.Spherical |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology Function |
| open scoped ENNReal RealInnerProductSpace |
| namespace PettyProjection.Spherical |
| open Scalar |
|
|
| lemma scalar_expectation {n : ℕ} (hn : 4 ≤ n) [NeZero n] |
| (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) : |
| mean (fun u => L n (f u)) ≤ 2*mean (fun u => W n (f u)) ∧ |
| (mean (fun u => L n (f u)) = 2*mean (fun u => W n (f u)) ↔ ∀ u, f u = 1) := by |
| have hL := (continuous_L hn).comp f.continuous |
| have hW : Continuous (fun u => 2*W n (f u)) := continuous_const.mul (continuous_W_comp f hf) |
| have hp : ∀ u, L n (f u) ≤ 2*W n (f u) := fun u => (pointwise n hn (hf u)).1 |
| have hm := mean_mono (f := fun u => L n (f u)) (g := fun u => 2*W n (f u)) hL hW hp |
| rw [mean_const_mul] at hm |
| refine ⟨hm, ?_, ?_⟩ |
| · intro he |
| have hae : (fun u => L n (f u)) =ᵐ[sigma n] (fun u => 2*W n (f u)) := by |
| apply (integral_eq_iff_of_ae_le (continuous_integrable hL) (continuous_integrable hW) |
| (Eventually.of_forall hp)).mp |
| change mean (fun u => L n (f u)) = mean (fun u => 2*W n (f u)) |
| rw [mean_const_mul]; exact he |
| have heq := (hL.ae_eq_iff_eq (sigma n) hW).mp hae |
| intro u |
| exact (pointwise n hn (hf u)).2.mp (congrFun heq u) |
| · intro he |
| have heq : (fun u => L n (f u)) = (fun u => 2*W n (f u)) := by |
| funext u; exact (pointwise n hn (hf u)).2.mpr (he u) |
| rw [heq, mean_const_mul] |
|
|
| /-- The full arbitrary-norm estimate, including strictness. |
| It uses the actual spherical measure and the actual quadratic-removing Q. -/ |
| theorem norm_estimate {n : ℕ} (hn : 4 ≤ n) [NeZero n] |
| (g : Seminorm ℝ (Space n)) (hg : ∀ u : Sphere n, 0 < g u) |
| (hnorm : mean (fun u : Sphere n => (g u)⁻¹^n) = 1) : |
| coefficient n * ‖Q n (toH n (restrictContinuous (fun x => g x ^ exponent n) |
| ((seminorm_continuous n g).pow _)))‖^2 ≤ |
| (mean (fun u : Sphere n => (g u)^exponent n))^2 - |
| (mean (fun u : Sphere n => (g u)^(exponent n-1)))^2 ∧ |
| (coefficient n * ‖Q n (toH n (restrictContinuous (fun x => g x ^ exponent n) |
| ((seminorm_continuous n g).pow _)))‖^2 = |
| (mean (fun u : Sphere n => (g u)^exponent n))^2 - |
| (mean (fun u : Sphere n => (g u)^(exponent n-1)))^2 ↔ ∀ u : Sphere n, g u = 1) := by |
| let f := restrictContinuous g (seminorm_continuous n g) |
| let m := mean (fun u => (f u)^exponent n)-1 |
| let ell := mean (fun u => (f u)^(exponent n-1))-1 |
| let t := mean (fun u => G n (f u)) |
| obtain ⟨hm, hl, hlm, ht, htb⟩ := norm_mean_ranges hn f hg hnorm |
| have hcorr := scalar_correction hn hm ht htb |
| have hk : 0 < (exponent n : ℝ) := by unfold exponent; split_ifs <;> norm_num |
| have hkm : 0 ≤ (exponent n : ℝ)-1 := by unfold exponent; split_ifs <;> norm_num |
| have hrem := mean_square_remainder hk hm hl hkm hlm |
| have hred := norm_reduction hn g |
| have hred' : coefficient n * ‖Q n (toH n (restrictContinuous (fun x => g x^exponent n) |
| ((seminorm_continuous n g).pow _)))‖^2 ≤ |
| mean (fun u => L n (f u)) + m^2-ell^2 := by |
| have hd : mean (fun u : Sphere n => D n (g u)) = m-t := mean_D hn f |
| rw [hd] at hred |
| exact hred.trans (by dsimp only [m, ell, t, f, restrictContinuous, ContinuousMap.coe_mk] at *; linarith) |
| obtain ⟨hsc, hsce⟩ := scalar_expectation hn f hg |
| have hw : mean (fun u => W n (f u)) = m-ell := by |
| rw [mean_W_of_normalized f hg hnorm] |
| dsimp only [m, ell]; ring |
| have hend : 2*(m-ell)+m^2-ell^2 = |
| (mean (fun u : Sphere n => (g u)^exponent n))^2 - |
| (mean (fun u : Sphere n => (g u)^(exponent n-1)))^2 := by |
| dsimp only [m, ell, f, restrictContinuous, ContinuousMap.coe_mk]; ring |
| have hfinal : coefficient n * ‖Q n (toH n (restrictContinuous (fun x => g x^exponent n) |
| ((seminorm_continuous n g).pow _)))‖^2 ≤ |
| (mean (fun u : Sphere n => (g u)^exponent n))^2 - |
| (mean (fun u : Sphere n => (g u)^(exponent n-1)))^2 := |
| hred'.trans (by rw [← hend]; rw [hw] at hsc; linarith) |
| refine ⟨hfinal, ?_, ?_⟩ |
| · intro he |
| apply hsce.mp |
| rw [hw] |
| rw [← hend] at he |
| linarith |
| · intro he |
| have hpow : restrictContinuous (fun x => g x^exponent n) ((seminorm_continuous n g).pow _) = |
| ContinuousMap.const (Sphere n) 1 := by ext u; simp [restrictContinuous, he u] |
| rw [hpow, Q_constant, norm_zero, zero_pow (by norm_num : 2 ≠ 0), mul_zero] |
| have hem : (fun u : Sphere n => (g u)^exponent n) = (fun _ => (1 : ℝ)) := by |
| funext u; simp [he u] |
| have hel : (fun u : Sphere n => (g u)^(exponent n-1)) = (fun _ => (1 : ℝ)) := by |
| funext u; simp [he u] |
| rw [hem, hel, sub_self] |
|
|
| end PettyProjection.Spherical |
| end |
|
|
| end OAI |
|
|
