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12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import Mathlib
namespace OAI
noncomputable section
open Set Filter
open scoped Topology ZeroAtInfty BoundedContinuousFunction
namespace ParametrixDilation
variable {V : Type} [NormedAddCommGroup V] [NormedSpace ℝ V]
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
def dilMap (t : ℝ) : C(V,V) := ⟨dilHomeo t,(dilHomeo t).continuous⟩
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 _ _
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)
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
end
end OAI
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