File size: 3,463 Bytes
caec230 | 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 | import OAI.MeasureTheory.Falconer.Geometry.NonstationaryGeometry
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.Calculus.Deriv.Inv
namespace OAI
/-!
# A normalized affine line in a spherical chart
Differentiation in a gnomonic chart is the tangential projection of the
coordinate direction divided by the normalization. This connects the
failed perpendicular test to an ordinary one-variable phase derivative.
-/
namespace Falconer
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
noncomputable def normalizedLine (a t : E) (u : ℝ) : E :=
‖a + u • t‖⁻¹ • (a + u • t)
theorem normalizedLine_norm (a t : E) {u : ℝ} (hne : a + u • t ≠ 0) :
‖normalizedLine a t u‖ = 1 := by
rw [normalizedLine,norm_smul,Real.norm_eq_abs,
abs_of_pos (inv_pos.mpr (norm_pos_iff.mpr hne)),inv_mul_cancel₀ (norm_ne_zero_iff.mpr hne)]
private theorem norm_line_hasDerivAt (a t : E) {u : ℝ} (hne : a + u • t ≠ 0) :
HasDerivAt (fun x : ℝ => ‖a + x • t‖)
(inner ℝ (a + u • t) t / ‖a + u • t‖) u := by
have hz : HasDerivAt (fun x : ℝ => a + x • t) t u := by
simpa using ((hasDerivAt_id u).smul_const t).const_add a
have hsq := hz.norm_sq
have hh := hsq.sqrt (pow_ne_zero 2 (norm_ne_zero_iff.mpr hne))
have he : (2 * inner ℝ (a + u • t) t) / (2 * Real.sqrt (‖a+u•t‖^2)) =
inner ℝ (a+u•t) t / ‖a+u•t‖ := by
rw [Real.sqrt_sq (norm_nonneg _)]
ring
rw [he] at hh
simpa only [Real.sqrt_sq (norm_nonneg _)] using hh
theorem normalizedLine_hasDerivAt (a t : E) {u : ℝ} (hne : a + u • t ≠ 0) :
HasDerivAt (normalizedLine a t)
(‖a + u • t‖⁻¹ • perpendicularPart (normalizedLine a t u) t) u := by
have hz : HasDerivAt (fun x : ℝ => a + x • t) t u := by
simpa using ((hasDerivAt_id u).smul_const t).const_add a
have hi := (norm_line_hasDerivAt a t hne).inv (norm_ne_zero_iff.mpr hne)
have hh := hi.smul hz
convert hh using 1
· rfl
· simp only [normalizedLine,perpendicularPart,real_inner_smul_left,smul_sub,smul_smul,
div_eq_mul_inv,neg_mul]
module
theorem normalizedLine_phase_hasDerivAt (a t Δ : E) {u : ℝ} (hne : a + u • t ≠ 0) :
HasDerivAt (fun x => inner ℝ (normalizedLine a t x) Δ)
(‖a+u•t‖⁻¹ * inner ℝ (perpendicularPart (normalizedLine a t u) t) Δ) u := by
simpa only [inner_zero_right,add_zero,zero_add,real_inner_smul_left] using
(normalizedLine_hasDerivAt a t hne).inner ℝ (hasDerivAt_const u Δ)
/-- The normalization costs at most two in the failed-test lower bound. -/
theorem normalizedLine_phase_nonstationary (a t Δ : E) {u τ : ℝ}
(hne : a+u•t ≠ 0) (hnorm : ‖a+u•t‖ ≤ 2) (hτ : 0 ≤ τ)
(hderiv : τ/2 ≤ |inner ℝ (perpendicularPart (normalizedLine a t u) t) Δ|) :
τ/4 ≤ |deriv (fun x => inner ℝ (normalizedLine a t x) Δ) u| := by
rw [(normalizedLine_phase_hasDerivAt a t Δ hne).deriv,abs_mul,
abs_of_pos (inv_pos.mpr (norm_pos_iff.mpr hne))]
have hhalf : (1/2 : ℝ) ≤ ‖a+u•t‖⁻¹ := by
simpa only [one_div] using one_div_le_one_div_of_le (norm_pos_iff.mpr hne) hnorm
calc
τ/4 = (1/2 : ℝ)*(τ/2) := by ring
_ ≤ ‖a+u•t‖⁻¹ * |inner ℝ (perpendicularPart (normalizedLine a t u) t) Δ| :=
mul_le_mul hhalf hderiv (by positivity) (by positivity)
end Falconer
end OAI
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