| import OAI.Analysis.AncientFlat.Evolution.RowLinearity |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped BoundedContinuousFunction SchwartzMap |
| namespace SourceFrozenOrder |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| lemma exists_commutator_lipschitz (K : SchwartzMap V ℂ) (j : ℕ) (α β : ℝ) (hα : 0<α) : |
| ∃ C : ℝ, 0≤C ∧ ∀ (A : V → ℝ), Continuous A → (∀ y, A y∈Icc α β) → |
| ∀ (r q L : ℝ), 0<r → 0≤q → q≤r → 0≤L → |
| ∀ (a₀ a₁ f : V →ᵇ ℂ), |
| (∀ x y, ‖a₁ x-a₀ y‖≤L*(‖x-y‖+q)) → |
| (∀ x y, ‖a₁ x-a₁ y‖≤L*‖x-y‖) → ∀ x y, |
| ‖(a₁ x*value K A r f j x-value K A r (a₀*f) j x)- |
| (a₁ y*value K A r f j y-value K A r (a₀*f) j y)‖ ≤ |
| (r⁻¹)^j*‖f‖*L*C*‖x-y‖ := by |
| obtain ⟨N,hN,hNb⟩ := exists_norm_bound K j α β hα |
| obtain ⟨B,hB,hBb⟩ := exists_weighted_norm_bound K j α β hα |
| obtain ⟨D,hD,hDb⟩ := exists_weighted_norm_bound (SchwartzMap.fderivCLM ℝ V ℂ K) (j+1) α β hα |
| refine ⟨N+3*D+4*B,by positivity,fun A hA hAr r q L hr hq hqr hL a₀ a₁ f hc hl x y => ?_⟩ |
| let g (y : V) : V →ᵇ ℂ := a₁ y • f-a₀*f |
| have hg (w z u : V) : ‖g w u‖≤(L*‖f‖)*(‖z-u‖+(‖z-w‖+q)) := by |
| have he : g w u=(a₁ w-a₀ u)*f u := by simp only [g,BoundedContinuousFunction.coe_sub, |
| BoundedContinuousFunction.coe_smul,BoundedContinuousFunction.coe_mul,Pi.sub_apply,Pi.mul_apply,smul_eq_mul]; ring |
| rw [he,norm_mul] |
| have ht : ‖w-u‖≤‖z-u‖+‖z-w‖ := by |
| have hh := norm_sub_le_norm_sub_add_norm_sub w z u |
| rw [norm_sub_rev w z] at hh |
| linarith |
| calc |
| _ ≤ (L*(‖w-u‖+q))*‖f‖ := mul_le_mul (hc w u) (f.norm_coe_le_norm u) (norm_nonneg _) (by positivity) |
| _ ≤ _ := by nlinarith [mul_nonneg (mul_nonneg hL (norm_nonneg f)) (sub_nonneg.mpr ht)] |
| have he (w z : V) : value K A r (g w) j z=a₁ w*value K A r f j z-value K A r (a₀*f) j z := by |
| simpa only [smul_eq_mul] using value_smul_sub K j α β hα A hA hAr r hr f (a₀*f) (a₁ w) z |
| have ha (w : V) : ‖value K A r (g w) j w‖≤(r⁻¹)^j*(L*‖f‖)*(2*r)*B := by |
| have hb := hBb A hA hAr r hr (g w) w (L*‖f‖) q (by positivity) hq |
| (by simpa only [sub_self,norm_zero,zero_add] using hg w w) |
| exact hb.trans (by gcongr |
| by_cases hxy : ‖x-y‖≤r |
| · have hd (z : V) (hz : z∈Metric.closedBall y ‖x-y‖) : |
| ‖value (SchwartzMap.fderivCLM ℝ V ℂ K) A r (g y) (j+1) z‖≤ |
| (r⁻¹)^j*(L*‖f‖)*(3*D) := by |
| have hz' : ‖z-y‖≤‖x-y‖ := by simpa only [Metric.mem_closedBall,dist_eq_norm] using hz |
| have hb := hDb A hA hAr r hr (g y) z (L*‖f‖) (‖z-y‖+q) (by positivity) (by positivity) (hg y z) |
| calc |
| _ ≤ (r⁻¹)^(j+1)*(L*‖f‖)*(3*r)*D := hb.trans (by gcongr; linarith) |
| _ = _ := by rw [pow_succ]; field_simp |
| have hv : ‖value K A r (g y) j x-value K A r (g y) j y‖≤ |
| ((r⁻¹)^j*(L*‖f‖)*(3*D))*‖x-y‖ := by |
| exact (convex_closedBall y ‖x-y‖ : Convex ℝ _).norm_image_sub_le_of_norm_hasFDerivWithin_le |
| (fun z _ => (hasFDerivAt_value K A hA α β hα hAr r hr (g y) j z).hasFDerivWithinAt) |
| hd (Metric.mem_closedBall_self (norm_nonneg _)) (by simpa only [Metric.mem_closedBall,dist_eq_norm] using (le_rfl : ‖x-y‖≤‖x-y‖)) |
| calc |
| _ = ‖(a₁ x-a₁ y)*value K A r f j x+(value K A r (g y) j x-value K A r (g y) j y)‖ := by rw [he,he]; congr 1; ring |
| _ ≤ ‖a₁ x-a₁ y‖*‖value K A r f j x‖+‖value K A r (g y) j x-value K A r (g y) j y‖ := by simpa only [norm_mul] using norm_add_le ((a₁ x-a₁ y)*value K A r f j x) _ |
| _ ≤ (L*‖x-y‖)*((r⁻¹)^j*(‖f‖*N)) + ((r⁻¹)^j*(L*‖f‖)*(3*D))*‖x-y‖ := |
| add_le_add (mul_le_mul (hl x y) (hNb A hA hAr r hr f x) (norm_nonneg _) (by positivity)) hv |
| _ ≤ _ := by nlinarith [mul_nonneg (by positivity : 0≤(r⁻¹)^j*‖f‖*L*‖x-y‖) hB] |
| · have hrxy : r≤‖x-y‖ := le_of_not_ge hxy |
| rw [← he x x,← he y y] |
| calc |
| _ ≤ ‖value K A r (g x) j x‖+‖value K A r (g y) j y‖ := norm_sub_le .. |
| _ ≤ (r⁻¹)^j*(L*‖f‖)*(2*r)*B+(r⁻¹)^j*(L*‖f‖)*(2*r)*B := add_le_add (ha x) (ha y) |
| _ = (r⁻¹)^j*‖f‖*L*(4*B)*r := by ring |
| _ ≤ (r⁻¹)^j*‖f‖*L*(N+3*D+4*B)*‖x-y‖ := by gcongr; linarith |
| end SourceFrozenOrder |
| |
| end |
| |
| end OAI |
| |
