| import Mathlib |
| import OAI.Geometry.PrescribedRicci.FamilyJetNorm |
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| namespace OAI |
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| /-! Complex Jet. -/ |
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| section |
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| noncomputable section |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Classical BigOperators |
| namespace TameInterpolation |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| variable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ] [Nonempty κ] |
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| def cdir (v : E) (f : E → ℂ) (x : E) : ℂ := fderiv ℝ f x v |
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| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
| lemma cdir_smooth (v : E) {f : E → ℂ} (hf : ContDiff ℝ ∞ f) : ContDiff ℝ ∞ (cdir v f) := |
| (hf.fderiv_right (by simp)).clm_apply contDiff_const |
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| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
| lemma cdir_compact (v : E) {f : E → ℂ} (hf : HasCompactSupport f) : |
| HasCompactSupport (cdir v f) := |
| (hf.fderiv ℝ).comp_left (g:=fun L : E →L[ℝ] ℂ => L v) (by simp) |
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| def cjet (e : ι → E) (f : E → ℂ) : {j : ℕ} → (Fin j → ι) → E → ℂ |
| | 0,_ => f |
| | _j+1,w => cdir (e (w 0)) (cjet e f (w ∘ Fin.succ)) |
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| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| [Fintype ι] [Nonempty ι] in |
| lemma cjet_smooth (e : ι → E) {f : E → ℂ} (hf : ContDiff ℝ ∞ f) |
| {j : ℕ} (w : Fin j → ι) : ContDiff ℝ ∞ (cjet e f w) := by |
| induction j with |
| | zero => exact hf |
| | succ j ih => exact cdir_smooth _ (ih _) |
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| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| [Fintype ι] [Nonempty ι] in |
| lemma cjet_compact (e : ι → E) {f : E → ℂ} (hf : HasCompactSupport f) |
| {j : ℕ} (w : Fin j → ι) : HasCompactSupport (cjet e f w) := by |
| induction j with |
| | zero => exact hf |
| | succ j ih => exact cdir_compact _ (ih _) |
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| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| [Fintype ι] [Nonempty ι] in |
| lemma cjet_real_map (e : ι → E) {f : E → ℂ} (hf : ContDiff ℝ ∞ f) |
| (L : ℂ →L[ℝ] ℝ) {j : ℕ} (w : Fin j → ι) : |
| jet e (fun x => L (f x)) w = fun x => L (cjet e f w x) := by |
| induction j with |
| | zero => rfl |
| | succ j ih => |
| simp only [jet,cjet,ih] |
| funext x |
| dsimp only [dir,cdir] |
| exact congrArg (fun T : E →L[ℝ] ℝ => T (e (w 0))) |
| (L.hasFDerivAt.comp x ((cjet_smooth e hf _).differentiable (by simp) x).hasFDerivAt).fderiv |
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| def realComponents (f : κ → E → ℂ) : κ × Bool → E → ℝ := |
| fun a x => if a.2 then (f a.1 x).im else (f a.1 x).re |
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| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| [Fintype κ] [Nonempty κ] in |
| lemma realComponents_smooth (f : κ → E → ℂ) (hf : ∀ a, ContDiff ℝ ∞ (f a)) |
| (a : κ × Bool) : ContDiff ℝ ∞ (realComponents f a) := by |
| rcases a with ⟨a,b⟩; cases b |
| · exact Complex.reCLM.contDiff.comp (hf a) |
| · exact Complex.imCLM.contDiff.comp (hf a) |
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| omit [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] |
| [BorelSpace E] [Fintype κ] [Nonempty κ] in |
| lemma realComponents_compact (f : κ → E → ℂ) (hf : ∀ a, HasCompactSupport (f a)) |
| (a : κ × Bool) : HasCompactSupport (realComponents f a) := by |
| rcases a with ⟨a,b⟩; cases b |
| · exact (hf a).comp_left (g:=Complex.re) rfl |
| · exact (hf a).comp_left (g:=Complex.im) rfl |
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| lemma lpNorm_complex_le (f : E → ℂ) {p : ℝ≥0∞} (hp : 1 ≤ p) |
| (hr : MemLp (fun x => (f x).re) p volume) (hi : MemLp (fun x => (f x).im) p volume) : |
| lpNorm f p volume ≤ lpNorm (fun x => (f x).re) p volume + lpNorm (fun x => (f x).im) p volume := by |
| have hh := lpNorm_mono_real (hr.norm.add hi.norm) (f:=f) |
| (fun x => by simpa only [Real.norm_eq_abs,Pi.add_apply] using Complex.norm_le_abs_re_add_abs_im (f x)) |
| apply hh.trans |
| simpa only [lpNorm_norm hr.aestronglyMeasurable,lpNorm_norm hi.aestronglyMeasurable] using lpNorm_add_le hr.norm (g:=fun x => ‖(f x).im‖) hp |
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| lemma cjet_lpNorm_le (e : ι → E) (f : κ → E → ℂ) |
| (hf : ∀ a, ContDiff ℝ ∞ (f a)) (hc : ∀ a, HasCompactSupport (f a)) |
| (a : κ) {j : ℕ} (w : Fin j → ι) {p : ℝ≥0∞} (hp : 1 ≤ p) : |
| lpNorm (cjet e (f a) w) p volume ≤ 2*familyJetNorm e (realComponents f) j p := by |
| have hr : ContDiff ℝ ∞ (fun x => (cjet e (f a) w x).re) := Complex.reCLM.contDiff.comp (cjet_smooth e (hf a) w) |
| have hi : ContDiff ℝ ∞ (fun x => (cjet e (f a) w x).im) := Complex.imCLM.contDiff.comp (cjet_smooth e (hf a) w) |
| have hcr := (cjet_compact e (hc a) w).comp_left (g:=Complex.re) rfl |
| have hci := (cjet_compact e (hc a) w).comp_left (g:=Complex.im) rfl |
| have hl := lpNorm_complex_le _ hp (hr.continuous.memLp_of_hasCompactSupport hcr) (hi.continuous.memLp_of_hasCompactSupport hci) |
| have h1 := (lpNorm_jet_le e (realComponents f (a,false)) w p).trans (jetNorm_le_family e (realComponents f) (a,false) j p) |
| have h2 := (lpNorm_jet_le e (realComponents f (a,true)) w p).trans (jetNorm_le_family e (realComponents f) (a,true) j p) |
| change lpNorm (jet e (fun x => Complex.reCLM (f a x)) w) p volume ≤ _ at h1 |
| change lpNorm (jet e (fun x => Complex.imCLM (f a x)) w) p volume ≤ _ at h2 |
| rw [cjet_real_map e (hf a) Complex.reCLM] at h1 |
| rw [cjet_real_map e (hf a) Complex.imCLM] at h2 |
| change lpNorm (fun x => (cjet e (f a) w x).re) p volume ≤ _ at h1 |
| change lpNorm (fun x => (cjet e (f a) w x).im) p volume ≤ _ at h2 |
| linarith only [hl,h1,h2] |
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| end TameInterpolation |
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| end |
| end |
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| end OAI |
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