| import OAI.Geometry.RCD.Sobolev.BochnerSquareSaturation |
| import OAI.Geometry.RCD.Sobolev.FiniteDimensionalBochnerAE |
|
|
| namespace OAI |
|
|
| /-! # Exact frame completion from the scalar dimensional Bochner inequality -/ |
|
|
| open Finset Set Filter MeasureTheory TopologicalSpace |
| noncomputable section |
| namespace RegularPointCoordinates |
|
|
| theorem ae_finite_bochner_bound {X : Type*} [MeasurableSpace X] {m : ℕ} |
| (μ : Measure X) (B : X → Euclidean m →ₗ[ℝ] Euclidean m →ₗ[ℝ] ℝ) |
| (D : X → Euclidean m →ₗ[ℝ] ℝ) (N : ℝ) |
| (h : ∀ a, ∀ᵐ x ∂μ, (D x a)^2≤N*B x a a) : |
| ∀ᵐ x ∂μ, ∀ a, (D x a)^2≤N*B x a a := by |
| obtain ⟨S,hS,hdense⟩ := exists_countable_dense (Euclidean m) |
| let : Countable S := hS.to_subtype |
| filter_upwards [MeasureTheory.ae_all_iff.mpr (fun a : S => h a)] with x hx |
| have hc : IsClosed {a : Euclidean m | (D x a)^2≤N*B x a a} := |
| isClosed_le ((D x).toContinuousLinearMap.continuous.pow 2) |
| (continuous_const.mul (bilinear_diagonal_continuous (B x))) |
| have hall := closure_minimal (fun a ha => hx ⟨a,ha⟩) hc |
| rw [hdense.closure_eq] at hall |
| exact fun a => hall (Set.mem_univ a) |
|
|
| theorem dimensional_cross_equality {N r s : ℝ} (hN : 0<N) |
| (h : ∀ t : ℝ, (N+t*r)^2≤N*(N+2*t*s+t^2*r^2)) : r=s := by |
| have hq (t : ℝ) : 0≤(N*r^2-r^2)*(t*t)+(2*N*(s-r))*t+0 := by |
| have hh := h t |
| nlinarith only [hh] |
| have hd := discrim_le_zero hq |
| simp only [discrim,mul_zero,sub_zero] at hd |
| have hz : 2*N*(s-r)=0 := by nlinarith only [hd,sq_nonneg (2*N*(s-r))] |
| rcases mul_eq_zero.mp hz with ha | ha |
| · exact (ne_of_gt (mul_pos (by norm_num) hN) ha).elim |
| · linarith |
|
|
| theorem centered_square_bochner_pair {E : Type*} [AddCommGroup E] [Module ℝ E] |
| (B : E →ₗ[ℝ] E →ₗ[ℝ] ℝ) (v w zv zw : E) (f g H P : ℝ) |
| (hpair : B zv zw=2*g*B zv w+4*f*H+4*P^2) |
| (hlinear : B v zw=2*g*B v w+2*H) : |
| B (zv-(2*f) • v) (zw-(2*g) • w)=4*P^2 := by |
| simp only [map_sub,map_smul,LinearMap.sub_apply,LinearMap.smul_apply,smul_eq_mul] |
| rw [hpair,hlinear] |
| ring |
|
|
| theorem centered_square_source {E : Type*} [AddCommGroup E] [Module ℝ E] |
| (D : E →ₗ[ℝ] ℝ) (v z : E) (f G : ℝ) |
| (hz : D z=2*f*D v+2*G) : D (z-(2*f) • v)=2*G := by |
| simp only [map_sub,map_smul,smul_eq_mul,hz] |
| ring |
|
|
| theorem bochner_frame_completion {E : Type*} [AddCommGroup E] [Module ℝ E] |
| {n : ℕ} (hn : 0<n) (B : E →ₗ[ℝ] E →ₗ[ℝ] ℝ) (D : E →ₗ[ℝ] ℝ) |
| (hsym : ∀ a b, B a b=B b a) (hBE : ∀ a, (D a)^2≤(n:ℝ)*B a a) |
| (z : Fin n → E) (w : E) (r : ℝ) (a : Fin n → ℝ) |
| (hDz : ∀ i, D (z i)=2) |
| (hBz : ∀ i j, B (z i) (z j)=if i=j then 4 else 0) |
| (hDw : D w=2*r) (hBw : B w w=4*r^2) |
| (hcross : ∀ i, B (z i) w=4*(a i)^2) : r=∑ i, (a i)^2 := by |
| classical |
| let q := (1/2:ℝ) • ∑ i, z i |
| let v := (1/2:ℝ) • w |
| have hDq : D q=(n:ℝ) := by simp [q,map_sum,hDz]; ring |
| have hDv : D v=r := by simp [v,hDw] |
| have hBq : B q q=(n:ℝ) := by simp [q,map_sum,hBz]; ring |
| have hBv : B v v=r^2 := by simp [v,hBw]; ring |
| have hqv : B q v=∑ i, (a i)^2 := by |
| simp only [q,v,map_smul,map_sum,LinearMap.smul_apply,LinearMap.sum_apply,smul_eq_mul,hcross] |
| rw [← mul_sum] |
| ring |
| have hvq : B v q=∑ i, (a i)^2 := (hsym v q).trans hqv |
| apply dimensional_cross_equality (N := (n:ℝ)) (by exact_mod_cast hn) |
| intro t |
| have hh := hBE (q+t • v) |
| simp only [map_add,map_smul,LinearMap.add_apply,LinearMap.smul_apply,smul_eq_mul, |
| hDq,hDv,hBq,hBv,hqv,hvq] at hh |
| convert hh using 1 |
| ring |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
