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ef96466 | 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 | import Mathlib.Analysis.Calculus.ContDiff.Deriv
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
namespace OAI
/-! A C1 extension of classical radial data from the closed integration interval. -/
open Set
open scoped ContDiff
namespace DefocusingNLS
theorem spectralC1IntervalExtension (f : ℝ → ℂ) (R : ℝ) (hR : 0 ≤ R)
(hf : ∀ r ∈ Icc 0 R, DifferentiableAt ℝ f r)
(hd : ContinuousOn (deriv f) (Icc 0 R)) :
∃ F : ℝ → ℂ, ContDiff ℝ 1 F ∧ EqOn F f (Icc 0 R) := by
let c := fun r : ℝ => max 0 (min R r)
have hc (r : ℝ) : c r ∈ Icc 0 R := by
exact ⟨le_max_left _ _,max_le hR (min_le_left _ _)⟩
have hce (r : ℝ) (hr : r ∈ Icc 0 R) : c r=r := by
simp only [c,min_eq_right hr.2,max_eq_right hr.1]
let d := fun r => deriv f (c r)
have hdc : Continuous d := hd.comp_continuous
(continuous_const.max (continuous_const.min continuous_id)) hc
let F := fun r => f 0+∫ x in (0 : ℝ)..r, d x
have hF (r : ℝ) : HasDerivAt F (d r) r :=
(intervalIntegral.integral_hasDerivAt_right (hdc.intervalIntegrable 0 r)
hdc.aestronglyMeasurable.stronglyMeasurableAtFilter hdc.continuousAt).const_add (f 0)
have hFd : deriv F=d := funext (fun r => (hF r).deriv)
refine ⟨F,contDiff_one_iff_deriv.mpr
⟨fun r => (hF r).differentiableAt,by simpa only [hFd] using hdc⟩,fun r hr => ?_⟩
have hi : (∫ x in (0 : ℝ)..r, d x)=∫ x in (0 : ℝ)..r, deriv f x := by
apply intervalIntegral.integral_congr_Ioo_of_le hr.1
intro x hx
exact congrArg (deriv f) (hce x ⟨hx.1.le,hx.2.le.trans hr.2⟩)
have hj : (∫ x in (0 : ℝ)..r, deriv f x)=f r-f 0 := by
apply intervalIntegral.integral_eq_sub_of_hasDerivAt
· intro x hx
rw [uIcc_of_le hr.1] at hx
exact (hf x ⟨hx.1,hx.2.trans hr.2⟩).hasDerivAt
· exact (hd.mono (Icc_subset_Icc_right hr.2)).intervalIntegrable_of_Icc hr.1
change f 0+(∫ x in (0 : ℝ)..r, d x)=f r
rw [hi,hj]
abel
end DefocusingNLS
end OAI
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