openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralCanonicalParameterPower.lean
| 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 | |