| import OAI.Analysis.CircleDomains.Sobolev.SmoothCutoffLinePayment |
|
|
| namespace OAI |
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|
| /-! The actual smooth balloon cutoffs pay for all unit-speed affine |
| line segments with their indicator densities. -/ |
| open Set Function Filter MeasureTheory Metric |
| open scoped Topology ENNReal |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| theorem smooth_affine_line_increment {f g : ℂ → ℝ} |
| (hf : ContDiff ℝ 1 f) (hg : ∀ z, ‖fderiv ℝ f z‖ ≤ g z) |
| (z e : ℂ) (he : ‖e‖ ≤ 1) {x y : ℝ} |
| (hgi : IntervalIntegrable (fun t => g (z+t • e)) volume x y) : |
| |f (z+x • e)-f (z+y • e)| ≤ ∫ t in uIoc x y, g (z+t • e) := by |
| apply smooth_line_increment_payment (hf.comp (by fun_prop)) hgi |
| intro t _ |
| have hline : HasDerivAt (fun s : ℝ => z+s • e) e t := by |
| simpa using ((hasDerivAt_id t).smul_const e).const_add z |
| have hder := ((hf.differentiable_one (z+t • e)).hasFDerivAt.comp_hasDerivAt t hline).deriv |
| rw [hder,← Real.norm_eq_abs] |
| exact ((fderiv ℝ f (z+t • e)).unit_le_opNorm e he).trans (hg _) |
|
|
| theorem balloon_line_density_intervalIntegrable (c z e : ℂ) (d : ℝ) {C : ℝ} |
| (hC : 0 ≤ C) (x y : ℝ) : |
| IntervalIntegrable (fun t : ℝ => (ball c d).indicator (fun _ => C) (z+t • e)) volume x y := by |
| have hline : Continuous (fun t : ℝ => z+t • e) := by fun_prop |
| have hm : Measurable (fun t : ℝ => (ball c d).indicator (fun _ => C) (z+t • e)) := |
| (measurable_const.indicator (s := ball c d) measurableSet_ball).comp hline.measurable |
| apply intervalIntegrable_iff.mpr |
| apply (intervalIntegrable_const : IntervalIntegrable (fun _ : ℝ => C) volume x y).def'.mono' |
| hm.aestronglyMeasurable.restrict |
| apply Eventually.of_forall |
| intro t |
| simpa only [Real.norm_of_nonneg hC] using |
| norm_indicator_le_norm_self (s := ball c d) (fun _ : ℂ => C) (z+t • e) |
|
|
| theorem balloon_cutoff_line_payment {f : ℂ → ℝ} (hf : ContDiff ℝ 1 f) |
| {c : ℂ} {d C : ℝ} (hC : 0 ≤ C) (hd : 0 < d) |
| (hder : ∀ z, ‖fderiv ℝ f z‖ ≤ (ball c d).indicator (fun _ => C/d) z) |
| (z e : ℂ) (he : ‖e‖ ≤ 1) (x y : ℝ) : |
| |f (z+x • e)-f (z+y • e)| ≤ |
| ∫ t in uIoc x y, (ball c d).indicator (fun _ => C/d) (z+t • e) := |
| smooth_affine_line_increment hf hder z e he |
| (balloon_line_density_intervalIntegrable c z e d (div_nonneg hC hd.le) x y) |
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|
| end CircleDomainRigidity |
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| end |
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| end OAI |
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|
