File size: 6,346 Bytes
92416a2 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 | import Mathlib
namespace OAI
noncomputable section
namespace Problem340
section AngularFactor
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
theorem norm_fderiv_angular_power {θ : E → ℂ} {x : E}
(hθ : DifferentiableAt ℝ θ x) (hunit : ‖θ x‖ = 1) (k : ℕ) :
‖fderiv ℝ (fun y => θ y ^ k) x‖ = (k : ℝ) * ‖fderiv ℝ θ x‖ := by
rw [fderiv_fun_pow k hθ, norm_smul, nsmul_eq_mul, norm_mul, norm_pow, hunit]
simp
theorem norm_fderiv_cutoff_angular_power {χ θ : E → ℂ} {x : E}
(hχ : DifferentiableAt ℝ χ x) (hθ : DifferentiableAt ℝ θ x)
(hunit : ‖θ x‖ = 1) (k : ℕ) :
‖fderiv ℝ (fun y => χ y * θ y ^ k) x‖ ≤
‖fderiv ℝ χ x‖ + (k : ℝ) * ‖χ x‖ * ‖fderiv ℝ θ x‖ := by
rw [fderiv_fun_mul hχ (hθ.fun_pow k)]
calc
_ ≤ ‖χ x • fderiv ℝ (fun y => θ y ^ k) x‖ +
‖θ x ^ k • fderiv ℝ χ x‖ := norm_add_le _ _
_ = ‖fderiv ℝ χ x‖ + (k : ℝ) * ‖χ x‖ * ‖fderiv ℝ θ x‖ := by
rw [norm_smul, norm_smul, norm_pow, hunit,
norm_fderiv_angular_power hθ hunit k]
ring
theorem norm_fderiv_cutoff_angular_power_le {χ θ : E → ℂ} {x : E}
(hχ : DifferentiableAt ℝ χ x) (hθ : DifferentiableAt ℝ θ x)
(hunit : ‖θ x‖ = 1) (hcut : ‖χ x‖ ≤ 1)
{A B : ℝ} (hA : ‖fderiv ℝ χ x‖ ≤ A) (hB : ‖fderiv ℝ θ x‖ ≤ B)
(k : ℕ) :
‖fderiv ℝ (fun y => χ y * θ y ^ k) x‖ ≤ A + (k : ℝ) * B := by
calc
_ ≤ ‖fderiv ℝ χ x‖ + (k : ℝ) * ‖χ x‖ * ‖fderiv ℝ θ x‖ :=
norm_fderiv_cutoff_angular_power hχ hθ hunit k
_ ≤ A + (k : ℝ) * B := by
apply add_le_add hA
calc
_ ≤ (k : ℝ) * 1 * ‖fderiv ℝ θ x‖ :=
mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hcut (Nat.cast_nonneg k)) (norm_nonneg _)
_ ≤ (k : ℝ) * B := by
simpa using mul_le_mul_of_nonneg_left hB (Nat.cast_nonneg k)
omit [NormedAddCommGroup E] [NormedSpace ℝ E] in
theorem norm_cutoff_angular_power_le {χ θ : E → ℂ} {x : E}
(hunit : ‖θ x‖ = 1) (hcut : ‖χ x‖ ≤ 1) (k : ℕ) :
‖χ x * θ x ^ k‖ ≤ 1 := by
simpa [norm_mul, norm_pow, hunit] using hcut
end AngularFactor
def angularPhase (z : ℂ) : ℂ := ‖z‖⁻¹ • z
@[simp] theorem angularPhase_zero : angularPhase 0 = 0 := by
simp [angularPhase]
theorem norm_angularPhase {z : ℂ} (hz : z ≠ 0) : ‖angularPhase z‖ = 1 := by
rw [angularPhase, norm_smul, Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg z))]
exact inv_mul_cancel₀ (norm_ne_zero_iff.mpr hz)
theorem contDiffAt_angularPhase {z : ℂ} (hz : z ≠ 0) {n : WithTop ℕ∞} :
ContDiffAt ℝ n angularPhase z := by
exact ((contDiffAt_id.norm ℂ hz).inv (norm_ne_zero_iff.mpr hz)).smul contDiffAt_id
theorem norm_fderiv_angularPhase_le {z : ℂ} (hz : z ≠ 0) :
‖fderiv ℝ angularPhase z‖ ≤ 2 / ‖z‖ := by
have hn : DifferentiableAt ℝ (fun w : ℂ => ‖w‖) z :=
differentiableAt_id.norm ℂ hz
have hn0 : ‖z‖ ≠ 0 := norm_ne_zero_iff.mpr hz
have hinv := (hasDerivAt_inv hn0).comp_hasFDerivAt z hn.hasFDerivAt
have hderiv : fderiv ℝ angularPhase z =
‖z‖⁻¹ • ContinuousLinearMap.id ℝ ℂ +
((-(‖z‖ ^ 2)⁻¹) • fderiv ℝ (fun w : ℂ => ‖w‖) z).smulRight z := by
exact (hinv.smul (hasFDerivAt_id z)).fderiv
rw [hderiv]
calc
_ ≤ ‖‖z‖⁻¹ • ContinuousLinearMap.id ℝ ℂ‖ +
‖((-(‖z‖ ^ 2)⁻¹) • fderiv ℝ (fun w : ℂ => ‖w‖) z).smulRight z‖ :=
norm_add_le _ _
_ = 2 / ‖z‖ := by
rw [norm_smul, ContinuousLinearMap.norm_id,
ContinuousLinearMap.norm_smulRight_apply, norm_smul, norm_fderiv_norm hn,
norm_neg, Real.norm_of_nonneg (inv_nonneg.mpr (sq_nonneg ‖z‖)),
Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg z))]
field_simp
norm_num
theorem contDiff_cutoff_angular_power {χ : ℂ → ℂ} {n : WithTop ℕ∞}
(hχ : ContDiff ℝ n χ) (hzero : ∀ᶠ z in nhds (0 : ℂ), χ z = 0) (k : ℕ) :
ContDiff ℝ n (fun z => χ z * angularPhase z ^ k) := by
rw [contDiff_iff_contDiffAt]
intro z
by_cases hz : z = 0
· subst z
apply (contDiffAt_const : ContDiffAt ℝ n (fun _ : ℂ => (0 : ℂ)) 0).congr_of_eventuallyEq
filter_upwards [hzero] with z hz
simp [hz]
· exact hχ.contDiffAt.mul ((contDiffAt_angularPhase hz (n := n)).pow k)
theorem norm_fderiv_cutoff_phase_power_le {χ : ℂ → ℂ} {r A : ℝ}
(hr : 0 < r) (hχ : Differentiable ℝ χ)
(hzero : ∀ z, ‖z‖ < r → χ z = 0)
(hcut : ∀ z, ‖χ z‖ ≤ 1) (hA : ∀ z, ‖fderiv ℝ χ z‖ ≤ A)
(k : ℕ) (z : ℂ) :
‖fderiv ℝ (fun w => χ w * angularPhase w ^ k) z‖ ≤
A + (k : ℝ) * (2 / r) := by
have hA0 : 0 ≤ A := (norm_nonneg _).trans (hA z)
by_cases hz : ‖z‖ < r
· have heq : (fun w => χ w * angularPhase w ^ k) =ᶠ[nhds z] (fun _ => (0 : ℂ)) := by
filter_upwards [(isOpen_lt continuous_norm continuous_const).eventually_mem hz] with w hw
simp [hzero w hw]
rw [heq.fderiv_eq]
rw [fderiv_const_apply, norm_zero]
positivity
· have hzr : r ≤ ‖z‖ := le_of_not_gt hz
have hz0 : z ≠ 0 := norm_pos_iff.mp (hr.trans_le hzr)
apply norm_fderiv_cutoff_angular_power_le (hχ z)
((contDiffAt_angularPhase hz0 (n := 1)).differentiableAt one_ne_zero)
(norm_angularPhase hz0) (hcut z) (hA z)
exact (norm_fderiv_angularPhase_le hz0).trans
(div_le_div_of_nonneg_left (by norm_num) hr hzr)
theorem norm_fderiv_cutoff_phase_power_linear {χ : ℂ → ℂ} {r A : ℝ}
(hr : 0 < r) (hχ : Differentiable ℝ χ)
(hzero : ∀ z, ‖z‖ < r → χ z = 0)
(hcut : ∀ z, ‖χ z‖ ≤ 1) (hA : ∀ z, ‖fderiv ℝ χ z‖ ≤ A)
(k : ℕ) (z : ℂ) :
‖fderiv ℝ (fun w => χ w * angularPhase w ^ k) z‖ ≤
(A + 2 / r) * (1 + (k : ℝ)) := by
apply (norm_fderiv_cutoff_phase_power_le hr hχ hzero hcut hA k z).trans
have hA0 : 0 ≤ A := (norm_nonneg _).trans (hA z)
have hkir : 0 ≤ (k : ℝ) * A := mul_nonneg (Nat.cast_nonneg k) hA0
have hri : 0 < 2 / r := div_pos (by norm_num) hr
nlinarith
end Problem340
end
end OAI
|