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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 49 50 51 52 53 54 55 56 57 58 59 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralTailPowerBounds
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralCanonicalPhysicalParameter
namespace OAI
/-! Actual parameter derivatives of the outgoing values have the required radial powers. -/
open Set Filter Topology
open scoped ContDiff
namespace DefocusingNLS
local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ)
noncomputable def canonicalPhysicalParameterValues (ν : ℂ) (Y : ℂ → ℝ → E₄)
(z : ℂ) (r : ℝ) : ℂ × ℂ :=
spectralPhysicalValueMap
(deriv (fun lam => spectralPhysicalPair (ν-2*lam) (star ν-2*lam) (Y lam) r) z)
theorem canonicalPhysicalParameterValues_eq (ν η b : ℂ) (n : ℕ) (L : ℝ)
(c : ℂ × ℂ) (Y : ℂ → ℝ → E₄)
(hY : IsCanonicalHolomorphicColumn ν η b n L c Y) (z : ℂ)
(r : ℝ) (hr : 1<r) :
canonicalPhysicalParameterValues ν Y z r =
spectralPowerValues (ν-2*z) (star ν-2*z)
(fun t => (circularParameterShear t (Y z t) (deriv (fun lam => Y lam t) z)).1.1)
(fun t => (circularParameterShear t (Y z t) (deriv (fun lam => Y lam t) z)).2.1) r := by
have hp := spectralPhysicalPair_hasParameterDerivAt ν (star ν) z Y
(fun t => deriv (fun lam => Y lam t) z) r
(hY.2.2.1 z (Real.log r) (Real.log_pos hr).le).differentiableAt.hasDerivAt
rw [canonicalPhysicalParameterValues,hp.deriv]
rfl
theorem canonicalPhysicalParameterValues_derivative_bound (ν η b : ℂ) (n : ℕ) (hn : 1≤n)
(L : ℝ) (hX : HasRadialExterior ν n b L) (hb : b ≠ 0)
(c : ℂ × ℂ) (Y : ℂ → ℝ → E₄)
(hY : IsCanonicalHolomorphicColumn ν η b n L c Y) (z : ℂ)
(σ : ℝ) (hσ : 0<σ) (N : ℕ) :
∃ C : ℝ, 0≤C ∧ ∀ᶠ r in atTop,
‖iteratedDeriv N (canonicalPhysicalParameterValues ν Y z) r‖ ≤
C*r^(ν.re-2*z.re+σ-(N : ℝ)) := by
let f := fun t => (circularParameterShear t (Y z t) (deriv (fun lam => Y lam t) z)).1.1
let g := fun t => (circularParameterShear t (Y z t) (deriv (fun lam => Y lam t) z)).2.1
have hj := canonical_parameterShear_value_logJets ν η b n hn L hX hb c Y hY z σ hσ
have hsame : (ν-2*z).re=(star ν-2*z).re := by simp
obtain ⟨C,hC,hbd⟩ := spectralPowerValues_derivative_bound (ν-2*z) (star ν-2*z)
hsame f g σ hj.1 hj.2 N
refine ⟨C,hC,?_⟩
filter_upwards [hbd,eventually_gt_atTop (1 : ℝ)] with r hr hr₁
have heq : canonicalPhysicalParameterValues ν Y z =ᶠ[𝓝 r]
spectralPowerValues (ν-2*z) (star ν-2*z) f g := by
filter_upwards [Ioi_mem_nhds hr₁] with s hs
exact canonicalPhysicalParameterValues_eq ν η b n L c Y hY z s hs
rw [heq.iteratedDeriv_eq N]
simpa only [Complex.sub_re,Complex.mul_re,Complex.re_ofNat,Complex.im_ofNat,
zero_mul,sub_zero] using hr
end DefocusingNLS
end OAI
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