| import Mathlib |
|
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| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped Topology ZeroAtInfty BoundedContinuousFunction |
| namespace ParametrixDilation |
| variable {V : Type} [NormedAddCommGroup V] [NormedSpace ℝ V] |
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| def dilHomeo (t : ℝ) : V ≃ₜ V := Homeomorph.smulOfNeZero (Real.exp (-t/4)) (Real.exp_ne_zero _) |
| @[simp] lemma dilHomeo_apply (t : ℝ) (x : V) : dilHomeo t x=Real.exp (-t/4) • x := rfl |
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| def dilMap (t : ℝ) : C(V,V) := ⟨dilHomeo t,(dilHomeo t).continuous⟩ |
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| def dil (t : ℝ) : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ) := |
| (ZeroAtInftyContinuousMap.compLinearMap (R := ℂ) (dilHomeo (V := V) t).toCocompactMap).mkContinuous 1 |
| (fun f => by |
| change ‖f.toBCF.compContinuous (dilMap (V := V) t)‖≤1*‖f.toBCF‖ |
| simpa only [one_mul] using f.toBCF.norm_compContinuous_le (dilMap (V := V) t)) |
| @[simp] lemma dil_apply (t : ℝ) (f : C₀(V,ℂ)) (x : V) : |
| dil t f x=f (Real.exp (-t/4) • x) := rfl |
| lemma norm_dil_le (t : ℝ) (f : C₀(V,ℂ)) : ‖dil t f‖≤‖f‖ := by |
| change ‖f.toBCF.compContinuous (dilMap (V := V) t)‖≤‖f.toBCF‖ |
| exact f.toBCF.norm_compContinuous_le _ |
| lemma opNorm_dil_le (t : ℝ) : ‖dil (V := V) t‖≤1 := by |
| let : NormedAddCommGroup C₀(V,ℂ) := inferInstance |
| let : NormedSpace ℂ C₀(V,ℂ) := inferInstance |
| apply (dil (V := V) t).opNorm_le_bound zero_le_one |
| intro f |
| simpa only [one_mul] using norm_dil_le t f |
| @[simp] lemma dil_zero : dil (V := V) 0=ContinuousLinearMap.id ℂ C₀(V,ℂ) := by |
| ext f x |
| simp |
| lemma dil_add (s t : ℝ) : dil (V := V) (s+t)=(dil s).comp (dil t) := by |
| ext f x |
| simp only [dil_apply,ContinuousLinearMap.comp_apply] |
| rw [smul_smul,←Real.exp_add] |
| congr 3 |
| ring |
| lemma dil_comm (s t : ℝ) (f : C₀(V,ℂ)) : dil s (dil t f)=dil t (dil s f) := by |
| rw [←ContinuousLinearMap.comp_apply,←dil_add,add_comm, dil_add] |
| rfl |
| lemma norm_dil (t : ℝ) (f : C₀(V,ℂ)) : ‖dil t f‖=‖f‖ := by |
| apply le_antisymm (norm_dil_le t f) |
| calc |
| ‖f‖=‖dil (-t) (dil t f)‖ := by rw [←ContinuousLinearMap.comp_apply,←dil_add]; simp |
| _≤‖dil t f‖ := norm_dil_le _ _ |
|
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| lemma tends_dil_zero (f : C₀(V,ℂ)) : Tendsto (fun t : ℝ => dil t f) (𝓝 0) (𝓝 f) := by |
| apply Metric.tendsto_nhds.mpr |
| intro ε hε |
| obtain ⟨R,hR⟩ := Metric.closedBall_compl_subset_of_mem_cocompact |
| ((zero_at_infty f).eventually (Metric.ball_mem_nhds (0:ℂ) (by linarith : 0<ε/4))) (0:V) |
| let M : ℝ := max R 1 |
| have hM : 0<M := lt_of_lt_of_le (by norm_num : (0:ℝ)<1) (le_max_right _ _) |
| obtain ⟨δ,hδ,huniform⟩ := Metric.uniformContinuous_iff.mp (ZeroAtInftyContinuousMap.uniformContinuous f) (ε/2) (by linarith) |
| have hexp : Tendsto (fun t : ℝ => Real.exp (-t/4)) (𝓝 0) (𝓝 1) := by |
| convert (Real.continuous_exp.comp (continuous_neg.div_const 4)).tendsto (0:ℝ) using 1 <;> first | rfl | simp |
| have hevent := hexp.eventually (Metric.ball_mem_nhds (1:ℝ) |
| (lt_min (by norm_num : (0:ℝ)<1/2) (div_pos hδ (by positivity : 0<2*M+1)))) |
| filter_upwards [hevent] with t ht |
| have hc : |Real.exp (-t/4)-1|< min (1/2) (δ/(2*M+1)) := by simpa only [Metric.mem_ball,Real.dist_eq] using ht |
| have hc1 : (1:ℝ)/2<Real.exp (-t/4) := by |
| have := (abs_lt.mp (lt_of_lt_of_le hc (min_le_left _ _))).1 |
| linarith |
| have hcδ : |Real.exp (-t/4)-1| *(2*M)<δ := by |
| have hh := (lt_div_iff₀ (by positivity : 0<2*M+1)).mp (lt_of_lt_of_le hc (min_le_right _ _)) |
| nlinarith [abs_nonneg (Real.exp (-t/4)-1)] |
| have hb : dist (dil t f) f≤ε/2 := by |
| change dist (dil t f).toBCF f.toBCF≤ε/2 |
| apply (BoundedContinuousFunction.dist_le (by linarith)).mpr |
| intro x |
| change dist (f (Real.exp (-t/4) • x)) (f x)≤ε/2 |
| by_cases hx : ‖x‖≤2*M |
| · apply (huniform _).le |
| calc |
| dist (Real.exp (-t/4) • x) x=|Real.exp (-t/4)-1| *‖x‖ := by |
| rw [dist_eq_norm,show Real.exp (-t/4) • x-x=(Real.exp (-t/4)-1) • x by rw [sub_smul,one_smul],norm_smul,Real.norm_eq_abs] |
| _≤|Real.exp (-t/4)-1| *(2*M) := mul_le_mul_of_nonneg_left hx (abs_nonneg _) |
| _<δ := hcδ |
| · have hx' : R<‖x‖ := lt_of_le_of_lt (le_max_left R 1) (by linarith : M<‖x‖) |
| have hy' : R<‖Real.exp (-t/4) • x‖ := by |
| rw [norm_smul,Real.norm_of_nonneg (Real.exp_pos _).le] |
| have := le_max_left R 1 |
| nlinarith [norm_nonneg x] |
| have hfx : ‖f x‖<ε/4 := by |
| simpa only [Set.mem_ofPred_eq,Metric.mem_ball,dist_zero_right] using hR (by simpa only [mem_compl_iff,Metric.mem_closedBall,dist_zero_right,not_le] using hx') |
| have hfy : ‖f (Real.exp (-t/4) • x)‖<ε/4 := by |
| simpa only [Set.mem_ofPred_eq,Metric.mem_ball,dist_zero_right] using hR (by simpa only [mem_compl_iff,Metric.mem_closedBall,dist_zero_right,not_le] using hy') |
| calc |
| dist (f (Real.exp (-t/4) • x)) (f x)≤‖f (Real.exp (-t/4) • x)‖+‖f x‖ := dist_le_norm_add_norm _ _ |
| _≤ε/2 := by linarith |
| exact lt_of_le_of_lt hb (by linarith) |
|
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| lemma continuous_dil (f : C₀(V,ℂ)) : Continuous (fun t : ℝ => dil t f) := by |
| apply continuous_iff_continuousAt.mpr |
| intro t |
| have ht : Tendsto (fun r : ℝ => r-t) (𝓝 t) (𝓝 0) := by |
| convert ((continuous_id.sub continuous_const).tendsto t : Tendsto (fun r : ℝ => r-t) (𝓝 t) (𝓝 (t-t))) using 1 <;> first | rfl | simp |
| have h := (tends_dil_zero (dil t f)).comp ht |
| change Tendsto (fun r => dil r f) (𝓝 t) (𝓝 (dil t f)) |
| convert h using 1 |
| ext r x |
| change dil r f x=dil (r-t) (dil t f) x |
| rw [←ContinuousLinearMap.comp_apply,←dil_add,sub_add_cancel] |
| end ParametrixDilation |
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| end |
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| end OAI |
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