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2fca478 | 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 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import Mathlib
import OAI.Geometry.PrescribedRicci.FamilyJetNorm
namespace OAI
/-! Complex Jet. -/
section
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]
def cdir (v : E) (f : E → ℂ) (x : E) : ℂ := fderiv ℝ f x v
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
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)
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))
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 _)
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 _)
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
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
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)
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
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
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]
end TameInterpolation
end
end
end OAI
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