| 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 |
|
|
