openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralCanonicalParameterL2.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralCanonicalParameterPower | |
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralTailWeightedL2 | |
| namespace OAI | |
| /-! The differentiated canonical outgoing values have finite weighted top energy. -/ | |
| open Set Filter Topology MeasureTheory | |
| open scoped ContDiff | |
| namespace DefocusingNLS | |
| local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ) | |
| theorem canonicalPhysicalParameterValues_top_integrable (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) | |
| (hs : 11+2*(ν.re-2*z.re+σ-(N : ℝ)) < -1) : | |
| ∃ T : ℝ, 1<T ∧ IntegrableOn | |
| (fun r => r^11*‖iteratedDeriv N (canonicalPhysicalParameterValues ν Y z) r‖^2) (Ioi T) := 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 | |
| have hs' : 11+2*((ν-2*z).re+σ-(N : ℝ)) < -1 := by | |
| simpa only [Complex.sub_re,Complex.mul_re,Complex.re_ofNat,Complex.im_ofNat, | |
| zero_mul,sub_zero] using hs | |
| obtain ⟨T,hT,hi⟩ := spectralPowerValues_top_integrable (ν-2*z) (star ν-2*z) | |
| hsame f g σ hj.1 hj.2 N hs' | |
| refine ⟨T,hT,hi.congr_fun ?_ measurableSet_Ioi⟩ | |
| intro r hr | |
| have hr₁ : 1<r := hT.trans hr | |
| 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 | |
| dsimp only | |
| rw [heq.iteratedDeriv_eq N] | |
| end DefocusingNLS | |
| end OAI | |